Chapter 9

Multiple Integrals and Line and Surface Integrals

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Section 9.1Introduction[34/100%]

Problem 3463

Determine the limits of integration in the integral

\iint \limits _{S} f(x,y) \, dx \, dy,

where $S$ is the disk $x^{2}+y^{2}=a^{2}$ , if the integration is performed first with respect to $x$ and then with respect to $y$ .

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Problem 3464

The same question as in the previous problem, for the area $S$ bounded by the lines $x=0$ , $y=0$ , $x+y=a$ .

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Problem 3465

In the integral

\iint \limits _{S} f(x,y) \, dx \, dy

the area $S$ is bounded by the lines $x=y$ , $y=0$ , $x+y=2$ . Determine the limits of integration if the inner integration is performed with respect to $x$ , and also if it is performed with respect to $y$ .

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Problem 3466

Determine the limits of integration in the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ if $S$ is the parallelogram with sides $y=0$ ; $y=a$ ; $y=x$ ; $y=x-2a$ . Consider both possible orders of integration.

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Problem 3467

Determine the limits of integration in the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ if $S$ is the area bounded by the lines $y=h$ ; $ay=hx$ ; $ay=4ah-hx$ , where $a>0$ ; $h>0$ .

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Problem 3468

Determine the limits of integration in the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ if $S$ is the region between the line $y=x-2$ and the parabola $y^{2}=x$ (both ways).

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Problem 3469

Determine the limits of integration in the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ if $S$ is the parallelogram with sides $y=x$ ; $y=x+a$ ; $y=2x$ ; $y=2x-a$ (both ways).

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Problem 3470

Determine the limits of integration in the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ if $S$ is the region given by the inequalities $x^{2}-y^{2} \leqslant a^{2}$ ; $x^{2}+y^{2} \leqslant 3a^{2}$ .

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Problem 3471

Reverse the order of integration in the integral

\int \limits _{0}^{1} \left(\int \limits _{0}^{x} f(x,y) \, dy\right) dx.

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Problem 3472

Reverse the order of integration in the integral

\int \limits _{0}^{2} dx \int \limits _{-\sqrt{1-(x-1)^{2}}}^{2} f(x,y) \, dy.

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Problem 3473

Reverse the order of integration in the integral

\int \limits _{0}^{1} dx \int \limits _{-\sqrt{1-y^{2}}}^{1-y} f(x,y) \, dy.

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Problem 3474

Reverse the order of integration in the integral

\int \limits _{0}^{a} dx \int \limits _{\frac{a^{2}-x^{2}}{2a}}^{\sqrt{a^{2}-x^{2}}} f(x,y) \, dy.

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Problem 3475

Reverse the order of integration in the integral

\int \limits _{0}^{2a} \int \limits _{\sqrt{2ax-x^{2}}}^{\sqrt{2ax}} f(x,y) \, dy \, dx.

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Problem 3476

Reverse the order of integration in the integral

\int \limits _{0}^{2} \int \limits _{5-\sqrt{25-x^{2}}}^{x} f(x,y) \, dy \, dx + \int \limits _{2}^{4} \int \limits _{5-\sqrt{25-x^{2}}}^{2} f(x,y) \, dy \, dx

(and combine).

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Problem 3477

Reverse the order of integration, and also pass to polar coordinates, in the integral

\int \limits _{0}^{a} \int \limits _{a-y}^{\sqrt{a^{2}-ay}} f(x,y) \, dx \, dy.

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Problem 3478

Reverse the order of integration, and also pass to polar coordinates, in the integral

\int \limits _{a}^{2a} \int \limits _{2a-y}^{\sqrt{2ay-y^{2}}} f(x,y) \, dx \, dy.

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Problem 3479

Determine the limits of integration in the double integral $\displaystyle \iint \limits_{S} f(x,y) \, ds$ in Cartesian and polar coordinates, if the region is defined as follows: $x^{2}+y^{2} \leqslant 4x$ ; $y \geqslant x$ .

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Problem 3480

Determine the limits of integration in the double integral $\displaystyle \iint \limits_{S} f(x,y) \, ds$ in Cartesian and polar coordinates, if the region is defined as follows: $x^{2}+y^{2} \leqslant a^{2}$ ; $x+y \geqslant a$ .

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Problem 3481

Determine the limits of integration in the double integral $\displaystyle \iint \limits_{S} f(x,y) \, ds$ in Cartesian and polar coordinates, if the region is defined as follows: $x \geqslant y$ ; $x+y \leqslant 2a$ ; $y \geqslant 0$ .

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Problem 3482

Determine the limits of integration in the double integral $\displaystyle \iint \limits_{S} f(x,y) \, ds$ in Cartesian and polar coordinates, if the region is defined as follows: $y^{2} \leqslant 2px+p^{2}$ ; $y \geqslant x$ .

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Problem 3483

Determine the limits of integration in the double integral $\displaystyle \iint \limits_{S} f(x,y) \, ds$ in Cartesian and polar coordinates, if the region is defined as follows: $x^{2}+y^{2} \geqslant 2a^{2}$ ; $x^{2}+y^{2} \leqslant 2ax$ .

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Problem 3484

Introduce new variables in place of $x$ and $y$ and determine the limits of integration with respect to the new variables for the integral $\displaystyle \int_{0}^{a} \int_{\alpha x}^{\beta x} f(x,y) \, dx \, dy$ , if $u=x+y$ , $uv=y$ .

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Problem 3485

Introduce new variables in place of $x$ and $y$ and determine the limits of integration with respect to the new variables for the integral $\displaystyle \int_{0}^{a} dx \int_{0}^{b} f(x,y) \, dy$ , if $u=y+\alpha x$ , $uv=y$ .

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Problem 3486

Introduce new variables in place of $x$ and $y$ and determine the limits of integration with respect to the new variables for the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ , if $x=r\cos^{3}\varphi$ , $y=r\sin^{3}\varphi$ , and the area $S$ is bounded by the astroid $x^{2/3}+y^{2/3}=a^{2/3}$ .

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Problem 3487

In the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ the area $S$ is bounded by the lines $y=\alpha x$ , $y=\beta x$ , $x=a$ (a triangle). By a suitable change of variables turn the region of integration into a rectangle. Explain how a smooth change of variables leads to the appearance of extraneous corners.

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Problem 3488

In the integral $\displaystyle \iint \limits_{S} f(x,y) \, dx \, dy$ the region of integration is a quarter-disk: $x^{2}+y^{2}=a^{2}$ , $x>0$ , $y>0$ . By a change of variables turn it into a rectangle.

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Problem 3489

In the same integral as in the previous problem (the region is a quarter-disk $x^{2}+y^{2}=a^{2}$ , $x>0$ , $y>0$ ), turn the region of integration into an isosceles right triangle.

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Problem 3490

Compute the double integral $\displaystyle \iint \limits_{S}(x+y) \, dx \, dy$ . The boundary of the region $S$ : $x=0$ , $y=0$ , $x+y=2$ .

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Problem 3491

Compute the double integral $\displaystyle \iint \limits_{S}(x^{2}+y^{2}) \, dx \, dy$ . The boundary of the region $S$ : $y=0$ , $x=y$ , $x=1$ .

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Problem 3492

Compute the double integral $\displaystyle \iint \limits_{S}(x+y) \, dx \, dy$ . The boundary of the region $S$ : $x=y$ , $x=y^{2}$ .

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Problem 3493

Compute the double integral

\iint \limits _{\left|x\right|+\left|y\right|<1} x^{2} \, dx \, dy.

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Problem 3494

Compute the double integral $\displaystyle \iint \limits_{S}(x-y) \, dx \, dy$ . The boundary of the region $S$ : $y=0$ , $x=y$ , $x+y=2$ .

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Problem 3495

Compute the double integral $\displaystyle \iint \limits_{S} xy \, dx \, dy$ . The boundary of the region $S$ : $y^{2}=2x$ , $x=2$ .

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Problem 3496

Compute the integrals, after first reversing the order of integration:

$\displaystyle \int \limits_{0}^{1} \int \limits_{\sqrt{y}}^{1} \frac{1}{x} e^{-y/x} \, dx \, dy$ ;
$\displaystyle \int \limits_{0}^{a} \int \limits_{x}^{a} e^{y^{2}} \, dy \, dx$ .

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Section 9.2Computing Areas[36/94%]

Problem 3497

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ , compute the area bounded by the curves $y=x^{2}$ , $x=y^{2}$ .

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Problem 3498

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ , compute the area bounded by the curves $y=2x-x^{2}$ , $y=x^{2}$ .

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Problem 3499

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ , compute the area bounded by the curves $2y=x^{2}$ , $x=y$ .

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Problem 3500

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ , compute the area bounded by the curves $4y=x^{2}-4x$ , $x-y-3=0$ .

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Problem 3501

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ and passing to polar coordinates, compute the area bounded by the lemniscate $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})$ and the circle $x^{2}+y^{2}=2ax$ .

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Problem 3502

Using the formula $\displaystyle S=\iint \limits_{S} dx \, dy$ and passing to polar coordinates, compute the area bounded by the cardioid $(x^{2}+y^{2}-ax)^{2}=a^{2}(x^{2}+y^{2})$ and the circle $x^{2}+y^{2}=ay\sqrt{3}$ .

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Problem 3503

Introducing polar coordinates, compute the area bounded by the lemniscate $(x^{2}+y^{2})^{2}=2a^{2}xy$ .

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Problem 3504

Introducing polar coordinates, compute the area bounded by the pedal curve of the ellipse $(x^{2}+y^{2})^{2}=a^{2}x^{2}+b^{2}y^{2}$ .

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Problem 3505

Introducing polar coordinates, compute the area bounded by the curve $(x^{2}+y^{2})^{3}=a^{2}(x^{4}+y^{4})$ .

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Problem 3506

Introducing polar coordinates, compute the area bounded by the curve $(x^{2}+y^{2})^{2}=a^{2}x^{2}-b^{2}y^{2}$ .

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Problem 3507

Introducing polar coordinates, compute the area of the loop of the folium of Descartes $x^{3}+y^{3}=axy$ .

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Problem 3508

Introducing polar coordinates, compute the area bounded by the curve $(x^{2}+y^{2})^{2}=a(x^{3}+y^{3})$ .

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Problem 3509

Introducing polar coordinates, compute the area bounded by the curve $x^{4}+y^{4}=2a^{2}xy$ .

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Problem 3510

Let $\omega (x,y)$ , $\varphi (x,y)$ , $\psi (x,y)$ be homogeneous functions of degrees $n$ , $n-2$ , $n-2$ , which vanish simultaneously only at $x=0$ and $y=0$ and are positive for other values of $x$ and $y$ . Denote by $S(a,b)$ the area bounded by the curve

\omega (x,y)=a\varphi (x,y)+b\psi (x,y), \qquad a \geqslant 0, \quad b \geqslant 0.

Prove that $S(a+\alpha , b+\beta )=S(a,b)+S(\alpha ,\beta )$ .

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Problem 3511

In particular, if $\omega (x,y)=(x^{2}+y^{2})^{2}$ , $\varphi (x,y)=x^{2}$ , $\psi (x,y)=y^{2}$ , then $S(a,b)=S(b,a)$ . Deduce from this that $S(a,b)=\dfrac {\pi }{2}(a^{2}+b^{2})$ (see problem 3510: here $S(a,b)$ is the area bounded by the curve $\omega (x,y)=a\varphi (x,y)+b\psi (x,y)$ for homogeneous functions $\omega$ , $\varphi$ , $\psi$ of degrees $n$ , $n-2$ , $n-2$ , which vanish simultaneously only at $x=0$ , $y=0$ and are positive at all other points, with $a \geqslant 0$ , $b \geqslant 0$ ).

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Problem 3512

In the following problems it is useful to apply generalized polar coordinates, setting $x=ar\cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\varphi$ , where $a$ , $b$ , and $\sigma$ are suitably chosen constants; the Jacobian of this substitution equals $J=ab\sigma r\cos^{\sigma -1}\varphi \sin^{\sigma -1}\varphi$ .

Find the area bounded by the curve

\frac{x^{4}}{a^{4}}+\frac{y^{4}}{b^{4}}=\frac{x^{2}}{h^{2}}+\frac{y^{2}}{k^{2}}.

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Problem 3513

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}=\dfrac {x}{a}-\dfrac {y}{b}$ ; $y>0$ .

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Problem 3514

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{3}=\dfrac {x^{2}}{h^{2}}$ ; $y>0$ , $a>0$ , $b>0$ .

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Problem 3515

Find the area of the loop of the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{3}=\dfrac {xy}{c^{2}}$ .

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Problem 3516

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$ ; $x>0$ , $y>0$ .

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Problem 3517

Find the area bounded by the curve $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {x^{2}}{h^{2}}-\dfrac {y^{2}}{k^{2}}$ ; $x>0$ , $y>0$ .

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Problem 3518

Find the area bounded by the curve $\left(\left(\dfrac {x}{a}\right)^{2/3}+\left(\dfrac {y}{b}\right)^{2/3}\right)^{6}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$ .

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Problem 3519

Find the area bounded by the curve $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{8}=\dfrac {x^{2}}{h^{2}}+\dfrac {y^{2}}{k^{2}}$ ; $x>0$ , $y>0$ .

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Problem 3520

Find the area of the loop of the curve $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{8}=\dfrac {xy}{c^{2}}$ .

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Problem 3521

In the following problems the boundary of the area is given by equations of the form $\varphi (x,y,a)=0$ , $\varphi (x,y,b)=0$ , $\psi (x,y,\alpha )=0$ , $\psi (x,y,\beta )=0$ . In these it is convenient to express $x$ and $y$ in terms of new variables $u$ and $v$ from the equations $\varphi (x,y,u)=0$ , $\psi (x,y,v)=0$ . Having found the Jacobian $J$ , we can compute $S$ by the formula $\displaystyle S=\iint \limits_{S} dx \, dy=\int \limits_{a}^{b} du \int \limits_{\alpha }^{\beta } \left|J\right| \, dv$ .

Find the area bounded by the curves $x+y=a$ , $x+y=b$ , $y=\alpha x$ , $y=\beta x$ ; $a<b$ , $\alpha <\beta$ .

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Problem 3522

Find the area bounded by the curves $x^{2}=ay$ , $x^{2}=by$ , $y=m$ , $y=n$ ; $a<b$ , $m<n$ .

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Problem 3523

Find the area bounded by the curves $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}=1$ , $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}=2$ , $\dfrac {x}{a}=\dfrac {y}{b}$ , $\dfrac {x}{a}=9\dfrac {y}{b}$ ; $x>0$ , $y>0$ .

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Problem 3524

Find the area bounded by the curves $xy=a^{2}$ , $xy=b^{2}$ , $y=m$ , $y=n$ ; $a^{2}>b^{2}$ , $m>n$ .

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Problem 3525

Find the area bounded by the curves $y^{2}=mx$ , $y^{2}=nx$ , $x=\alpha y$ , $x=\beta y$ ; $m>n$ , $\alpha >\beta$ .

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Problem 3526

Find the area bounded by the curves $xy=a^{2}$ , $xy=b^{2}$ , $x=\alpha y$ , $x=\beta y$ ; $x>0$ , $y>0$ , $a>b$ , $\alpha >\beta$ .

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Problem 3527

Find the area bounded by the curves $y^{2}=ax$ , $y^{2}=bx$ , $x^{2}=my$ , $x^{2}=ny$ ; $a>b$ , $m>n$ .

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Problem 3528

Find the area bounded by the curves $xy=a^{2}$ , $xy=b^{2}$ , $y^{2}=mx$ , $y^{2}=nx$ ; $a^{2}>b^{2}$ , $m>n$ .

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Problem 3529

Find the area bounded by the curves $y^{2}=a^{2}-2ax$ , $y^{2}=b^{2}-2bx$ , $y^{2}=m^{2}+2mx$ , $y^{2}=n^{2}+2nx$ ; $y>0$ , $a>b$ , $m>n$ .

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Problem 3530

Find the area bounded by the curves

\frac{x^{2}}{\operatorname {ch}^{2}u_{0}}+\frac{y^{2}}{\operatorname {sh}^{2}u_{0}}=c^{2}, \qquad \frac{x^{2}}{\operatorname {ch}^{2}u_{1}}+\frac{y^{2}}{\operatorname {sh}^{2}u_{1}}=c^{2}; \qquad u_{1}>u_{0}, \quad x>0;

\frac{x^{2}}{\cos ^{2}v_{0}}-\frac{y^{2}}{\sin ^{2}v_{0}}=c^{2}, \qquad \frac{x^{2}}{\cos ^{2}v_{1}}-\frac{y^{2}}{\sin ^{2}v_{1}}=c^{2}; \qquad v_{1}>v_{0}, \quad y>0.

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Remark.

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The curves bounding the region are confocal ellipses and hyperbolas intersecting at a right angle. It is useful here to introduce elliptic coordinates $u$ and $v$ by setting $x=c\operatorname {ch}u\cos v$ , $y=c\operatorname {sh}u\sin v$ .

Problem 3531

Find the area bounded by the ellipse $(ax+by+c)^{2}+(a_{1}x+b_{1}y+c_{1})^{2}=h^{2}$ ; $ab_{1}-a_{1}b \neq 0$ .

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Problem 3532

Find the area bounded by the line $\left|ax+by+c\right|+\left|a_{1}x+b_{1}y+c_{1}\right|=h$ ; $ab_{1}-a_{1}b \neq 0$ .

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Section 9.3Computing Volumes[56/93%]

Problem 3533

Find the volume bounded by the surfaces $x+y+z=6$ , $x=0$ , $z=0$ , $x+2y=4$ .

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Problem 3534

Find the volume bounded by the surfaces $x-y+z=6$ , $x+y=2$ , $x=y$ , $y=0$ , $z=0$ .

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Problem 3535

Find the volume bounded by the surfaces $x+y+z=6$ , $x=y$ , $y=0$ , $x=3$ , $z=0$ .

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Problem 3536

Find the volume bounded by the surfaces $z=a+x$ , $z=-x-a$ , $x^{2}+y^{2}=a^{2}$ .

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Problem 3537

Find the volume bounded by the surfaces $z^{2}=xy$ , $x=a$ , $y=b$ , $z=0$ .

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Problem 3538

Find the volume bounded by the surfaces $x+y+z=a$ , $3x+y=a$ , $3x+2y=2a$ , $y=0$ , $z=0$ .

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Problem 3539

Find the volume bounded by the surfaces $x+y+z=a$ , $x^{2}+y^{2}=b^{2}$ , $z=0$ ; $a>b\sqrt{2}$ .

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Problem 3540

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , $y=\dfrac {b}{a}x$ , $y=0$ , $z=0$ ; $x>0$ .

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Problem 3541

Find the volume bounded by the surface $x^{2}z^{2}+a^{2}y^{2}=c^{2}x^{2}$ , $0<x<a$ .

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Problem 3542

Find the volume bounded by the surfaces $y^{2}+z^{2}=x$ , $x=y$ ; $z>0$ .

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Problem 3543

Find the volume bounded by the surfaces $z=x^{2}+y^{2}$ , $y=x^{2}$ ; $0<y<1$ , $z=0$ .

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Problem 3544

Find the volume bounded by the surfaces $z=\cos x \cos y$ ; $\left|x+y\right|<\dfrac {\pi }{2}$ , $z=0$ .

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Problem 3545

Find the volume bounded by the surfaces $z=\sin (x^{2}+y^{2})$ , $z=0$ ; $n\pi <x^{2}+y^{2}<(n+1)\pi$ .

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Problem 3546

Find the volume bounded by the surfaces $x^{2}+y^{2}=cz$ , $x^{2}+y^{2}=ax$ , $z=0$ .

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Problem 3547

Find the volume bounded by the surfaces $x^{2}+y^{2}=2ax$ , $z=\alpha z$ , $z=\beta x$ ; $\alpha <\beta$ .

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Remark.

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The plane surface " $z=\alpha z$ " is reproduced verbatim from the source's own printed text (by sense it should be a family of planes tilted with respect to the $x$ -axis, analogous to $z=\beta x$ ); this may be a typo in the original.

Problem 3548

Find the volume bounded by the surfaces $x^{2}+y^{2}=\alpha z^{2}$ , $x^{2}+y^{2}=ax$ ; $z>0$ .

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Problem 3549

Find the volume bounded by the surfaces $x^{2}+y^{2}=az$ , $x+z=2a$ .

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Problem 3550

Find the volume bounded by the surfaces $cz=xy$ , $x^{2}+y^{2}=ax$ , $z=0$ ; $y>0$ .

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Problem 3551

Find the volume bounded by the surfaces $z=x^{2}+y^{2}$ , $z=x+y$ .

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Problem 3552

Prove that the volume bounded by the plane $z=mx+ny+p$ and the paraboloid $z=x^{2}+y^{2}$ equals half the area of the circle $x^{2}+y^{2}=mx+cy+p$ , multiplied by the difference of the quantities $x^{2}+y^{2}$ and $mx+cy+p$ , taken at the center of that circle.

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Problem 3553

Prove that the volume of the segment cut off by a plane from an elliptic paraboloid equals the area of the base of the segment, multiplied by half the height of the segment.

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Problem 3554

A part is cut from a circular cone with base radius $R$ and height $H$ by a plane parallel to the axis of the cone. Determine the volume of the cut-off part, if the segment at the base corresponds to a central angle $2\alpha$ .

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Problem 3555

A right prism with a square base (of side $a$ ) and height $H$ is intersected by a circular cone in such a way that its trace on one base is the circle inscribed in the base, and on the other base is the circle circumscribed about it. What fraction of the volume of the prism lies outside the cone?

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Problem 3556

A right hexagonal prism (a nut) with face width $a$ is intersected by a coaxial conical surface whose generator makes an angle $\alpha =60^{\circ }$ with the axis. Determine the volume between the lateral surface of the prism, its base, and the cone, if the cone intersects the base along the circle inscribed in the base. In the following problems, as in some of the preceding ones, it is useful to introduce polar coordinates.

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Problem 3557

Find the volume bounded by the surfaces $x^{2}+y^{2}=cz$ , $x^{4}+y^{4}=a^{2}(x^{2}+y^{2})$ , $z=0$ .

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Problem 3558

Find the volume bounded by the surfaces $z^{2}=2xy$ , $(x^{2}+y^{2})^{2}=2a^{2}xy$ , $z=0$ ; $x>0$ , $y>0$ .

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Problem 3559

Find the volume bounded by the surfaces $x^{2}+y^{2}+z^{2}=a^{2}$ , $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ .

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Problem 3560

Find the volume bounded by the surfaces $x^{2}+y^{2}+z^{2}=a^{2}$ , $x^{2}+y^{2}>\left|ax\right|$ . (Viviani's problem.)

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Problem 3561

Find the volume bounded by the surfaces $z=x^{2}+y^{2}$ , $z^{2}=xy$ ; $x>0$ , $y>0$ .

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Problem 3562

Find the volume bounded by the surfaces $z(x+y)=ax+by$ , $z=0$ ; $1<x^{2}+y^{2}<4$ , $x>0$ , $y>0$ , $a>0$ , $b>0$ .

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Problem 3563

Prove that the volume of the body bounded by the surfaces:

z=0, \qquad x^{2}+y^{2}=c^{2}, \qquad z(\varphi (x)+\varphi (y))=a\varphi (x)+b\varphi (y),

where $\varphi (x)$ is any positive integrable function, $a>0$ and $b>0$ , equals $\dfrac {1}{2}\pi c^{2}(a+b)$ .

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Problem 3564

Prove that the volume bounded by the surfaces $z=0$ and $z=e^{-x^{2}-y^{2}}$ equals $\pi$ .

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Problem 3565

By a linear change of variables, find the volume enclosed between the surfaces $z=0$ and $z=Ae^{-(ax^{2}+bxy+cy^{2})}$ , where $a>0$ , $b^{2}-4ac<0$ .

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Problem 3566

Find the volume bounded by the surfaces $\dfrac {x^{2}}{p}+\dfrac {y^{2}}{q}=2z$ , $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ , $z=0$ .

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Problem 3567

Find the volume bounded by the surfaces $cz=xy$ , $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ , $z=0$ ; $x>0$ , $y>0$ .

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Problem 3568

Find the volume bounded by the surface $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{2}+\dfrac {z^{2}}{c^{2}}=1$ .

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Problem 3569

Find the volume bounded by the surface $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{k}+\dfrac {z}{c}=1$ ; $z>0$ ; $k>0$ .

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Problem 3570

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z^{2}}{c^{2}}$ , $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {x}{h}$ ; $z>0$ .

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Problem 3571

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z}{c}$ , $\dfrac {x^{4}}{a^{4}}+\dfrac {y^{4}}{b^{4}}=\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}$ , $z=0$ .

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Problem 3572

Find the volume bounded by the surfaces $z^{2}=2xy$ , $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{2}=\dfrac {2xy}{c^{2}}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3573

Find the volume bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z}{c}$ , $\dfrac {x}{a}+\dfrac {z}{c}=2$ .

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Problem 3574

Find the volume bounded by the surface $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}+\dfrac {z^{2}}{c^{2}}=1$ ; $x>0$ , $y>0$ , $z>0$ .

?

Problem 3575

Find the volume bounded by the surface $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}+\dfrac {z^{2}}{c^{2}}=1$ ; $x>0$ , $y>0$ , $z>0$ .

?

Problem 3576

Find the volume bounded by the surfaces $z=x\sqrt{x}+y\sqrt{y}$ , $x+y=1$ ; $x>0$ , $y>0$ , $z>0$ .

?

Problem 3577

Find the volume bounded by the surfaces $z^{2}=xy$ , $x+y=1$ ; $z>0$ .

?

Problem 3578

Find the volume bounded by the surfaces $cz=xy$ , $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {xy}{ab}$ ; $x>0$ , $y>0$ , $z>0$ .

?

Problem 3579

Find the volume bounded by the surfaces $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}+\dfrac {z^{2}}{c^{2}}=1$ , $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}=\dfrac {x}{a}$ ; $y>0$ , $z>0$ .

?

Problem 3580

Find the volume bounded by the surface

\left(\left(\frac{x}{a}\right)^{2/3}+\left(\frac{y}{b}\right)^{2/3}\right)^{3}+\left(\frac{z}{c}\right)^{2}=1.

?

Problem 3581

Find the volume bounded by the surface $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{8}+\left(\dfrac {z}{c}\right)^{2}=1$ ; $x>0$ , $y>0$ , $z>0$ .

?

Problem 3582

Find the volume bounded by the surface $\left(\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}\right)^{4}+\left(\dfrac {z}{c}\right)^{2}=1$ ; $x>0$ , $y>0$ , $z>0$ .

?

Problem 3583

Find the volume bounded by the surfaces $z=ye^{-\frac{xy}{a^{2}}}$ , $xy=a^{2}$ , $xy=2a^{2}$ , $y=m$ , $y=n$ , $z=0$ .

?

Remark.

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In method of solution, this problem and those up to and including 3588 are close to problems 3521--3530 of this same section.

Problem 3584

Find the volume bounded by the surfaces $cz=xy$ , $y^{2}=mx$ , $y^{2}=nx$ , $x=\alpha y$ , $x=\beta y$ , $z=0$ .

?

Problem 3585

Find the volume bounded by the surfaces $z^{2}=xy$ , $xy=a^{2}$ , $xy=4a^{2}$ , $x=2y$ , $x=3y$ , $z=0$ .

?

Problem 3586

Find the volume bounded by the surfaces $cz=xy$ , $y^{2}=2x$ , $y^{2}=3x$ , $x^{2}=y$ , $x^{2}=2y$ , $z=0$ .

?

Problem 3587

Find the volume bounded by the surfaces $z^{2}=xy$ , $xy=1$ , $xy=4$ , $y^{2}=x$ , $y^{2}=3x$ , $z=0$ .

?

Problem 3588

Find the volume bounded by the surfaces $z=x^{2}y$ , $y^{2}=a^{2}-2ax$ , $y^{2}=m^{2}+2mx$ , $y=0$ , $z=0$ .

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Section 9.4Computing Surfaces[28/100%]

Problem 3589

The area of a surface given by the equation $F(x,y,z)=0$ is expressed by the integral

\iint \limits _{S} \frac{\sqrt{\left(\dfrac {\partial F}{\partial x}\right)^{2}+\left(\dfrac {\partial F}{\partial y}\right)^{2}+\left(\dfrac {\partial F}{\partial z}\right)^{2}}}{\left|\dfrac {\partial F}{\partial z}\right|} \, dx \, dy,

where the area $S$ is the projection of the required part of the surface onto the plane $xOy$ . If the surface is given by the parametric equations $x=\varphi (u,v)$ , $y=\psi (u,v)$ , $z=\omega (u,v)$ , and the arc-length element $ds$ of a curve lying on the surface is expressed by the formula $ds^{2}=E\, du^{2}+2F\, du\, dv+G\, dv^{2}$ , where

E=\left(\frac{\partial x}{\partial u}\right)^{2}+\left(\frac{\partial y}{\partial u}\right)^{2}+\left(\frac{\partial z}{\partial u}\right)^{2}, \qquad G=\left(\frac{\partial x}{\partial v}\right)^{2}+\left(\frac{\partial y}{\partial v}\right)^{2}+\left(\frac{\partial z}{\partial v}\right)^{2},

F=\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}+\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}+\frac{\partial z}{\partial u}\frac{\partial z}{\partial v},

then the area of the surface is represented by the integral $\displaystyle \iint \sqrt{EG-F^{2}} \, du \, dv$ , extended over the region of values of $u$ and $v$ corresponding to the points of the part of the surface under study.

Find the area of the surface $z^{2}=2xy$ for $z>0$ , $0<x<a$ , $0<y<b$ .

?

Problem 3590

Find the area of the surface $x^{2}=2pz$ for $\alpha x>y>\beta x$ , $0<x<a$ .

?

Problem 3591

Find the area of the part of the surface $z^{2}=2px$ cut out by the surfaces $y^{2}=2qx$ , $x=a$ .

?

Problem 3592

Find the area of the part of the cylinder $\dfrac {x^{2}}{a^{2}}+\dfrac {z^{2}}{b^{2}}=1$ lying inside the cylinder $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{c^{2}}=1$ ; $a>b$ .

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Problem 3593

Find the area of the part of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ lying inside the cylinder $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ ; $a>b$ .

?

Problem 3594

Find the area of the part of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ lying outside the cylinders $x^{2}+y^{2}=\pm ax$ . (Viviani's problem.)

?

Problem 3595

Find the area of the part of the cylinders $x^{2}+y^{2}=\pm ax$ lying inside the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ .

?

Problem 3596

Find the area of the part of the cone $y^{2}+z^{2}=x^{2}$ lying inside the cylinder $x^{2}+y^{2}=a^{2}$ .

?

Problem 3597

Find the area of the part of the same cone $y^{2}+z^{2}=x^{2}$ cut out by the surface $x^{2}=ay$ .

?

Problem 3598

Find the area of the part of the surface $z=\operatorname {arctg}\dfrac {y}{x}$ whose projection onto the plane $xOy$ is bounded by the first turn of the Archimedean spiral $r=\varphi$ and the axis $Ox$ .

?

Problem 3599

Find the area of the surface $2cz=y^{2}-x^{2}+2xy\operatorname {ctg}\alpha$ under the conditions $z \geqslant 0$ ; $x^{2}+y^{2}<a^{2}$ .

?

Problem 3600

Find the area of the part of the surface $x^{2}+y^{2}=2az$ inside the cylinder $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ .

?

Problem 3601

Find the area of the part of the surface $az=xy$ inside the cylinder $(x^{2}+y^{2})^{2}=2a^{2}xy$ .

?

Problem 3602

Find the area of the part of the surface $x^{2}+y^{2}+z^{2}=a^{2}$ inside the cylinder $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ .

?

Problem 3603

Find the area of the part of the surface $x^{2}+y^{2}=z^{2}$ inside the cylinder $(x^{2}+y^{2})^{2}=2xy$ for $z \geqslant 0$ .

?

Problem 3604

Find the area of the part of the surface $(x\cos \alpha +y\sin \alpha )^{2}+z^{2}=a^{2}$ for $x>0$ , $y>0$ , $z>0$ .

?

Problem 3605

Find the area of the part of the surface $(x+y)^{2}+z=1$ for $x>0$ , $y>0$ , $z>0$ .

?

Problem 3606

Find the area of the part of the surface $(x+y)^{2}+2z^{2}=2a^{2}$ for $x>0$ , $y>0$ , $z>0$ .

?

Problem 3607

Find the area of the part of the surface $(x^{2}+y^{2})z=x+y$ for $1<x^{2}+y^{2}<4$ , $x>0$ , $y>0$ .

?

Problem 3608

Find the area of the part of the surface $\dfrac {x^{2}}{a}+\dfrac {y^{2}}{b}=2z$ inside the cylinder $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ .

?

Problem 3609

Find the area of the part of the surface $\dfrac {x^{2}}{a}-\dfrac {y^{2}}{b}=2z$ inside the cylinder $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ for $z>0$ .

?

Problem 3610

Find the area of the part of the surface $\dfrac {x^{2}}{a}+\dfrac {y^{2}}{b}=2z$ inside the cylinder $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{2}=\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}$ .

?

Problem 3611

Find the area of the part of the surface $x^{2}+y^{2}+z^{2}=a^{2}$ for $0<x$ , $0<y$ , $x+y<a$ , $z>0$ .

?

Problem 3612

Find the area of the part of the surface $z^{2}=2xy$ for $0<x$ , $0<y$ , $0<z$ , $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}<1$ .

?

Problem 3613

Find the area of the part of the surface $\sin z=\operatorname {sh}x \operatorname {sh}y$ for $1<x<2$ .

?

Problem 3614

Find the area of the part of the helicoid $x=r\cos \varphi$ , $y=r\sin \varphi$ , $z=h\varphi$ for $0<r<a$ , $0<\varphi <2\pi$ .

?

Problem 3615

Find the area of the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}+y^{2})$ .

?

Problem 3616

Find the solid angle at which the rectangle $0<y<b$ , $0<z<c$ , $x=a>0$ is seen from the origin. In other words, find the part of the surface of the sphere $x^{2}+y^{2}+z^{2}=1$ onto which the points of the given rectangle are projected by rays from the origin.

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Section 9.5Line Integrals[17/94%]

Problem 3617

If the equation of a curve is given either as a single equation $f(x,y)=0$ , or parametrically: $x=\varphi (t)$ , $y=\psi (t)$ , then the line integral along the curve $AB$ , denoted by $\displaystyle \int_{AB} P(x,y) \, dx+Q(x,y) \, dy$ , is evaluated by reducing it to an ordinary definite integral: if $x=\varphi (t)$ , $y=\psi (t)$ and the parameter $t$ varies from $a$ to $b$ as we pass from $A$ to $B$ , then

\int _{AB} P \, dx+Q \, dy=\int _{a}^{b} \big(P(\varphi ,\psi )\varphi '+Q(\varphi ,\psi )\psi '\big) \, dt.

When the direction of the path of integration is reversed, the value of the integral is multiplied by minus one: $\displaystyle \int_{AB} P \, dx+Q \, dy=-\int_{BA} P \, dx+Q \, dy$ .

Find the value of the integral $\displaystyle \int_{AB} x \, dy-y \, dx$ along the parabola $y=x^{2}$ between the points $A(0,0)$ and $B(2,4)$ .

?

Problem 3618

Find the integral $\displaystyle \int_{AB}(x+y) \, dx+(x-y) \, dy$ , if the path of integration is the parabola $y=x^{2}$ between the points $(-1,1)$ , $(1,1)$ .

?

Problem 3619

Find the integral $\displaystyle \int_{AB} x^{2} \, ds$ , where the path of integration is the upper half of the circle $x^{2}+y^{2}=a^{2}$ between the points $A(a,0)$ and $B(-a,0)$ .

?

Problem 3620

Find the value of the integral $\displaystyle \int_{AB} x \, dy$ , taken along the right half-circle $x^{2}+y^{2}=a^{2}$ between the points $A(0,-a)$ and $B(0,a)$ .

?

Problem 3621

Find the value of the integral $\displaystyle \int_{AB}(x^{2}+y^{2}) \, dx+(x^{2}-y^{2}) \, dy$ , taken around the contour of the triangle with vertices at the points $A(0,0)$ , $B(1,0)$ , $C(0,1)$ .

?

Remark.

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Green's formula: $\displaystyle \int_{C} P \, dx+Q \, dy=\iint \limits_{S}\left(\dfrac {\partial Q}{\partial x}-\dfrac {\partial P}{\partial y}\right) dx \, dy$ transforms the line integral taken around a closed contour $C$ in the positive direction of traversal (i.e., counterclockwise) into an integral over the area $S$ bounded by that contour. It follows that the integral $\int_{AB} P \, dx+Q \, dy$ , taken between the points $A$ and $B$ along two different paths, has the same value if, in the region enclosed between these paths, the functions $P$ and $Q$ satisfy $\dfrac {\partial Q}{\partial x}=\dfrac {\partial P}{\partial y}$ .

Problem 3622

From Green's formula (see problem 3621) derive the equality: $S=\dfrac {1}{2}\int_{C}(x \, dy-y \, dx)$ , where $C$ is a contour enclosing the area $S$ in the positive direction.

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Problem 3623

Using the formula of problem 3622 ( $S=\dfrac {1}{2}\int_{C}(x\, dy-y\, dx)$ ), compute the area of the ellipse $x=a\cos t$ , $y=b\sin t$ .

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Problem 3624

Using the formula of problem 3622 ( $S=\dfrac {1}{2}\int_{C}(x\, dy-y\, dx)$ ), compute the area of the sector of the hyperbola $x=a\operatorname {ch}t$ , $y=b\operatorname {sh}t$ , bounded by the axis $Ox$ , the arc of the hyperbola up to the point $M(x_{0},y_{0})$ , and the line $OM$ .

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Problem 3625

Using the formula of problem 3622 ( $S=\dfrac {1}{2}\int_{C}(x\, dy-y\, dx)$ ), compute the area of the sector of the astroid $x=a\cos^{3}t$ , $y=a\sin^{3}t$ , bounded by the lines $OM_{1}$ and $OM_{2}$ from the origin and the arc $M_{1}M_{2}$ of the astroid (the points $M_{1}$ , $M_{2}$ correspond to the parameter values $t_{1}$ , $t_{2}$ ).

?

Problem 3626

Find the area of the three-cusped hypocycloid:

x=a(2\cos t+\cos 2t), \qquad y=a(2\sin t-\sin 2t).

?

Problem 3627

Find the value of the line integral $\displaystyle \int_{C} \dfrac {x \, dy-y \, dx}{x^{2}+y^{2}}$ , taken around a closed contour $C$ in the positive direction.

?

Problem 3628

The curves $X(x,y)=0$ and $Y(x,y)=0$ have several simple points of intersection. Compute the value of the integral $\dfrac {1}{2\pi }\displaystyle \int_{C} \dfrac {X\, dY-Y\, dX}{X^{2}+Y^{2}}$ , taken in the positive direction around a closed contour $C$ .

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Problem 3629

One application of line integrals is finding the center of gravity $(x_{c}, y_{c}, z_{c})$ of a curve. For this the following formulas are used:

Mx_{c}=\int _{AB} x\mu \, ds, \qquad My_{c}=\int _{AB} y\mu \, ds, \qquad Mz_{c}=\int _{AB} z\mu \, ds, \qquad M=\int _{AB} \mu \, ds,

where $\mu (x,y,z)$ is the density of the curve at the point $(x,y,z)$ , and $M$ is the mass of the curve.

Find the coordinates of the center of gravity of a homogeneous arc of a circle of radius $a$ subtending a central angle $2\varphi$ .

?

Problem 3630

Find the coordinates of the center of gravity of the entire arc of the homogeneous cardioid $r=a(1+\cos \varphi )$ .

?

Problem 3631

Find the coordinates of the center of gravity of one arch of the homogeneous cycloid

x=a(t-\sin t), \qquad y=a(1-\cos t) \qquad \text{for } 0<t<2\pi .

?

Problem 3632

Find the coordinates of the center of gravity of an arc of the helix

x=a\cos t, \qquad y=a\sin t, \qquad z=ht

for $0<t<m$ .

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Problem 3633

Find the coordinates of the center of gravity of a homogeneous arc of the curve

x=e^{-t}\cos t, \qquad y=e^{-t}\sin t, \qquad z=e^{-t}

for $0<t<\infty$ .

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Section 9.6Some Applications of Double Integrals in Mechanics and Strength of Materials[27/89%]

Problem 3634

The coordinates of the center of gravity of a plate are found by the formulas:

Mx_{c}=\iint \limits _{S} x\mu \, dx \, dy, \qquad My_{c}=\iint \limits _{S} y\mu \, dx \, dy, \qquad M=\iint \limits _{S} \mu \, dx \, dy,

where $\mu =\mu (x,y)$ is the density of the plate at the point $(x,y)$ ; $M$ is the mass of the plate, and the quantities $Mx_{c}$ and $My_{c}$ are the static moments of the plate relative to the axes $Oy$ and $Ox$ .

Find the coordinates of the center of gravity of a homogeneous plate bounded by the lines $x=0$ , $y=0$ , $x+y=a$ .

?

Problem 3635

Find the coordinates of the center of gravity of a homogeneous plate bounded by the lines $y^{2}=2px$ , $y=0$ , $x=x_{0}$ .

?

Problem 3636

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{2/3}+y^{2/3}=a^{2/3}$ ; $x>0$ , $y>0$ .

?

Problem 3637

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{2}+y^{2}=a^{2}$ ; $x>0$ , $y>0$ .

?

Problem 3638

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{2}+y^{2}=a^{2}$ ; $\left|y\right|<\operatorname {tg}\alpha$ .

?

Remark.

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In the source the condition is given literally as $\left|y\right|<\operatorname {tg}\alpha$ (with no factor of $x$ ); the naturally expected condition for a sector is $\left|y\right|<x\operatorname {tg}\alpha$ , which is consistent with the answer.

Problem 3639

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $r^{2}=a^{2}\cos 2\varphi$ (the right-hand loop).

?

Problem 3640

Find the coordinates of the center of gravity of a homogeneous plate bounded by the cardioid $r=a(1+\cos \varphi )$ .

?

Problem 3641

Find the coordinates of the center of gravity of a homogeneous plate bounded by the loop of the curve $x^{3}+y^{3}=3axy$ .

?

Problem 3642

Find the coordinates of the center of gravity of a homogeneous plate bounded by the lines $x=a(t-\sin t)$ , $y=a(1-\cos t)$ ; $0<t<2\pi$ ; $y=0$ .

?

Problem 3643

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $x^{4}+y^{4}=x^{2}y$ (the right-hand loop).

?

Problem 3644

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $\sqrt{x}+\sqrt{y}=\sqrt{a}$ , $x=0$ , $y=0$ .

?

Problem 3645

Find the coordinates of the center of gravity of a homogeneous plate bounded by the line $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{4}=\dfrac {xy}{ab}$ (the right-hand loop).

?

Problem 3646

Prove that the volume of a right cylinder (or prism) cut off by a plane not necessarily parallel to the base equals the area of the base multiplied by the height above the center of gravity of the base.

?

Problem 3647

Prove that the volume of the solid obtained by rotating an area $S$ about an axis that does not intersect $S$ and lies in the same plane equals the area $S$ multiplied by the length of the circle traced by the center of gravity of the area $S$ . (Pappus's theorem, often called Guldin's theorem.)

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Problem 3648

The moment of inertia of an area $S$ relative to some axis lying in the same plane is called the integral $\displaystyle J=\iint \limits_{S} \delta^{2} \, dx \, dy$ , where $\delta$ is the distance from the variable point $(x,y)$ to the axis, and the integral is taken over the whole area $S$ . In particular, the moments of inertia relative to the axes $Ox$ and $Oy$ equal the integrals $J_{x}=\iint \limits_{S} y^{2} \, dx \, dy$ and $J_{y}=\iint \limits_{S} x^{2} \, dx \, dy$ . The polar moment of the area $S$ relative to some point is called the integral $\iint \limits_{S} r^{2} \, dx \, dy$ , where $r$ is the distance from the point $(x,y)$ to the given point. In particular, the polar moment relative to the origin equals $\iint \limits_{S}(x^{2}+y^{2}) \, dx \, dy$ . The centrifugal moment is called the integral $K_{xy}=\iint \limits_{S} xy \, dx \, dy$ . The curve $J_{x}x^{2}-2K_{xy}xy+J_{y}y^{2}=1$ is called the ellipse of inertia of the area $S$ .

Find the moments of inertia of a sector with radius $a$ and central angle $2\varphi$ relative to its axis of symmetry and relative to the perpendicular to it passing through the center of gravity.

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Problem 3649

Find the moment of inertia of a segment with radius $a$ and central angle $2\varphi$ relative to its axis of symmetry.

?

Problem 3650

Can the ellipse of inertia of a sector become a circle?

?

Problem 3651

An isosceles triangle has base $a$ and height $h$ . For what relation between $a$ and $h$ does the ellipse of inertia of the triangle relative to the center of gravity become a circle?

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Problem 3652

Find the moments of inertia of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ relative to the axes, and find the equation of the ellipse of inertia.

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Problem 3653

A ring between concentric circles of radii 13 and 12 has the same area as a circle of radius 5. How many times greater is the moment of inertia of the ring than that of the circle, if both moments are taken relative to a diameter?

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Problem 3654

Find the moment of inertia relative to the axis $Ox$ of the parallelogram with sides $a_{1}x+b_{1}y=\pm h_{1}$ , $a_{2}x+b_{2}y=\pm h_{2}$ .

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Problem 3655

Find the centrifugal moment of inertia of the same parallelogram (with sides $a_{1}x+b_{1}y=\pm h_{1}$ , $a_{2}x+b_{2}y=\pm h_{2}$ ).

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Problem 3656

Find the moment of inertia of the area of the curve $x^{4}+y^{4}=x^{2}+y^{2}$ relative to the axis $Ox$ and compare its value with the same moment of inertia of the square $\left|x+y\right|+\left|x-y\right|=2$ .

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Problem 3657

The torsional rigidity of a rod with a simply connected cross-section $S$ is measured by the quantity $C=2\mu \iint \limits_{S} \psi (x,y) \, dx \, dy$ , where $\mu$ is the shear modulus of the rod's material, and $\psi (x,y)$ is the stress function. The function $\psi (x,y)$ equals zero on the boundary of the cross-section, and inside it satisfies the equation $\dfrac {\partial^{2}\psi }{\partial x^{2}}+\dfrac {\partial^{2}\psi }{\partial y^{2}}=-2$ .

Find the rigidity $C$ for a circular cross-section, when $2\psi =a^{2}-x^{2}-y^{2}$ .

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Problem 3658

Find $C$ for an elliptical cross-section, when $(a^{2}+b^{2})\psi =a^{2}b^{2}-b^{2}x^{2}-a^{2}y^{2}$ , where $a$ and $b$ are the semi-axes of the ellipse.

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Problem 3659

The same question (find the torsional rigidity $C$ ) as in problem 3658, for an equilateral triangle with sides $x\pm y\sqrt{3}=h$ , $x=0$ . In this case $2h\psi =x(h-x-y\sqrt{3})(h-x+y\sqrt{3})$ .

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Problem 3660

The level of a liquid is the axis $Ox$ , and the axis $Oy$ is directed downward. Prove that the pressure of the liquid on a vertical plate $S$ equals $\gamma \iint \limits_{S} y \, dx \, dy$ , where $\gamma$ is the specific weight of the liquid. Prove that the point of application of the resultant of the pressure forces lies at a depth $h$ , where $\displaystyle h\iint \limits_{S} y \, dx \, dy=\iint \limits_{S} y^{2} \, dx \, dy$ .

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Section 9.7Surface Integrals, Coordinates of Centers of Gravity, and Moments of Inertia of Surfaces[14/93%]

Problem 3661

A surface integral is called an integral of the form $\displaystyle \iint \limits_{\sigma } P \, dy \, dz+Q \, dz \, dx+R \, dx \, dy$ . In computing it, the variables $x$ , $y$ , and $z$ are expressed in terms of two independent variables, using the equation of the surface $f(x,y,z)=0$ or its parametric equations $x=\varphi (u,v)$ , $y=\psi (u,v)$ , $z=\omega (u,v)$ . Another, more explicit way of writing surface integrals is $\displaystyle \iint \limits_{\sigma }(P\cos \alpha +Q\cos \beta +R\cos \gamma ) \, d\sigma$ , where $\alpha$ , $\beta$ , $\gamma$ are the angles of the normal with the coordinate axes; here $d\sigma =\dfrac {dx\, dy}{\cos \gamma }$ .

The coordinates of the centers of gravity of parts of homogeneous surfaces having mass are found using surface integrals by the formulas:

Sx_{c}=\iint \limits _{S} x \, ds, \qquad Sy_{c}=\iint \limits _{S} y \, ds, \qquad Sz_{c}=\iint \limits _{S} z \, ds; \qquad S=\iint \limits _{S} ds.

Here $S$ is the given part of the surface.

Find the coordinates of the center of gravity of the part of the shell of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ for $x>0$ , $y>0$ , $z>0$ .

?

Problem 3662

Find the coordinates of the center of gravity of the surface of a segment of the same sphere $x^{2}+y^{2}+z^{2}=a^{2}$ for $h<z<a$ .

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Problem 3663

Find the coordinates of the center of gravity of the surface of the helicoid $x=u\cos v$ , $y=u\sin v$ , $z=hv$ for $0<u<a$ , $0<v<\pi$ .

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Problem 3664

Find the coordinates of the center of gravity of the part of the surface $3z=2(x\sqrt{x}+y\sqrt{y})$ bounded by the planes $x=0$ , $y=0$ , $x+y=1$ .

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Problem 3665

Find the coordinates of the center of gravity of the surface $x^{2}=2c(c-z)$ , bounded by the surfaces: $y=kx$ , $y=0$ , $z=0$ .

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Problem 3666

Find the coordinates of the center of gravity of the part of the surface $x^{2}+y^{2}+z^{2}=a^{2}$ bounded by the surfaces: $x^{2}+y^{2}=ax$ , $z=0$ .

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Problem 3667

Find the coordinates of the center of gravity of the part of the surface $x^{2}+y^{2}+z^{2}=a^{2}$ bounded by the planes: $x+y=\pm a$ , $x-y=\pm a$ .

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Problem 3668

Find the moments of inertia of the part of the homogeneous surface of the cone $h^{2}(x^{2}+y^{2})=a^{2}z^{2}$ for $0<z<h$ relative to the axis $Oz$ .

?

Problem 3669

Find the moment of inertia of the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ relative to a diameter.

?

Problem 3670

Find the moment of inertia of the surface of the paraboloid $x^{2}+y^{2}=2az$ relative to $Oz$ for $0<z<a$ .

?

Problem 3671

Find the moment of inertia of the surface of the spherical segment $x^{2}+y^{2}+z^{2}=a^{2}$ for $h<z<a$ relative to $Oz$ .

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Problem 3672

Compute the double integral $\displaystyle \iint \dfrac {ds}{\rho }$ , taken over the surface of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , where $\rho$ is the distance from the center of the ellipsoid to the plane tangent to the element $ds$ of the surface of the ellipsoid.

?

Problem 3673

Compute the integral $\displaystyle \iint \dfrac {ds}{\rho^{n}}$ , taken over the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ , if $\rho$ is the distance from the surface element to the point $(0,0,c)$ , located outside the sphere.

?

Problem 3674

Show that the integral $\displaystyle w=\iint \limits_{S} \dfrac {\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}} \, ds=\iint \limits_{S} \dfrac {\partial \frac{1}{r}}{\partial n} \, ds$ , taken over the surface $S$ , equals the solid angle at which the surface $S$ is seen from the origin. Here $\overrightarrow {r}$ is the radius vector from the origin to the surface element $ds$ ; $\overrightarrow {n}$ is the normal to the surface, forming an acute angle with the vector $\overrightarrow {r}$ drawn from the origin. The quantity $\dfrac {\partial u}{\partial n}$ is defined by the equality

\frac{\partial u}{\partial n}=\frac{\partial u}{\partial x}\cos (\overrightarrow {n},x)+\frac{\partial u}{\partial y}\cos (\overrightarrow {n},y)+\frac{\partial u}{\partial z}\cos (\overrightarrow {n},z).

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Section 9.8The Triple Integral[7/100%]

Problem 3675

Determine the limits of integration in Cartesian, cylindrical, and spherical coordinates in the triple integral $\displaystyle \iiint \limits_{V} f(x,y,z) \, dv$ , taken over the volume $v$ given by the inequalities:

x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0; \qquad x+y \leqslant a; \qquad z \leqslant h.

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Problem 3676

Determine the limits of integration in Cartesian, cylindrical, and spherical coordinates in the triple integral $\displaystyle \iiint \limits_{V} f(x,y,z) \, dv$ , taken over the volume $v$ given by the inequalities:

x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0; \qquad x+y+z \leqslant a.

?

Problem 3677

Determine the limits of integration in Cartesian, cylindrical, and spherical coordinates in the triple integral $\displaystyle \iiint \limits_{V} f(x,y,z) \, dv$ , taken over the volume $v$ given by the inequalities:

x^{2}+y^{2}+z^{2} \leqslant a^{2}; \qquad x^{2}+y^{2} \leqslant z^{2}\operatorname {tg}^{2}\alpha .

?

Problem 3678

Determine the limits of integration in Cartesian, cylindrical, and spherical coordinates in the triple integral $\displaystyle \iiint \limits_{V} f(x,y,z) \, dv$ , taken over the volume $v$ given by the inequalities:

x^{2}+y^{2} \leqslant z^{2}; \qquad x^{2}+y^{2}+z^{2} \leqslant az.

?

Problem 3679

Determine the limits of integration in Cartesian, cylindrical, and spherical coordinates in the triple integral $\displaystyle \iiint \limits_{V} f(x,y,z) \, dv$ , taken over the volume $v$ given by the inequalities:

x^{2}+y^{2} \geqslant z^{2}; \qquad x^{2}+y^{2}+z^{2} \leqslant az.

?

Problem 3680

Determine the limits of integration in Cartesian, cylindrical, and spherical coordinates in the triple integral $\displaystyle \iiint \limits_{V} f(x,y,z) \, dv$ , taken over the volume $v$ given by the inequalities:

x^{2}+y^{2} \leqslant a^{2}; \qquad x+z \leqslant a; \qquad z \geqslant 0.

?

Problem 3681

Determine the limits of integration in Cartesian, cylindrical, and spherical coordinates in the triple integral $\displaystyle \iiint \limits_{V} f(x,y,z) \, dv$ , taken over the volume $v$ given by the inequalities:

x^{2}+y^{2} \leqslant z^{2}; \qquad x^{2}+z^{2} \leqslant a^{2}; \qquad z \geqslant 0.

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Section 9.9Computing Volumes[40/100%]

Problem 3682

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{2}=a^{3}x$ .

?

Problem 3683

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{2}=axyz$ .

?

Problem 3684

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{2}=az(x^{2}+y^{2})$ .

?

Problem 3685

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}+y^{2})$ .

?

Problem 3686

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}+y^{2}-z^{2})$ .

?

Problem 3687

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{3}=a^{2}y^{2}z^{2}$ .

?

Problem 3688

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{3}=a^{3}xyz$ .

?

Problem 3689

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{3}=a^{3}(x^{3}+y^{3}+z^{3})$ ; $x>0$ , $y>0$ , $z>0$ .

?

Problem 3690

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{5}=(a^{3}x^{2}+b^{3}y^{2}+c^{3}z^{2})^{2}$ .

?

Problem 3691

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{2}=a^{3}ze^{-\frac{x^{2}+y^{2}}{x^{2}+y^{2}+z^{2}}}$ .

?

Problem 3692

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2})^{2}+z^{4}=a^{3}(x-y)$ .

?

Problem 3693

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2})^{2}+z^{4}=a^{3}z$ .

?

Problem 3694

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2})^{3}+z^{6}=a^{3}xyz$ .

?

Problem 3695

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{3}=\dfrac {a^{6}z^{2}}{x^{2}+y^{2}}$ .

?

Problem 3696

Find the volume of the solid bounded by the given surface, introducing suitable new variables that simplify the integration (e.g., spherical coordinates): $(x^{2}+y^{2}+z^{2})^{2}=\dfrac {a^{6}}{x^{2}+y^{2}}$ .

?

Problem 3697

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}\Big)^{2}=\dfrac {x}{h}$ .

?

Problem 3698

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}\Big)^{2}=\dfrac {xyz}{h^{3}}$ .

?

Problem 3699

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}\Big)^{3}=\dfrac {z^{4}}{h^{4}}$ .

?

Problem 3700

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}\Big)^{3}=\dfrac {xyz}{h^{3}}$ .

?

Problem 3701

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}+\alpha^{2}\Big)^{2}=4\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\Big)$ ; $\alpha^{2}<1$ .

?

Problem 3702

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}\Big)^{2}=\dfrac {z}{h}e^{-\frac{z^{2}/c^{2}}{x^{2}/a^{2}+y^{2}/b^{2}+z^{2}/c^{2}}}$ .

?

Problem 3703

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\Big)^{2}+\dfrac {z^{4}}{c^{4}}=\dfrac {z}{k}$ .

?

Problem 3704

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates by the formulas $x=ar\sin \theta \cos \varphi$ , $y=br\sin \theta \sin \varphi$ , $z=cr\cos \theta$ (the Jacobian of this transformation is $J=abcr^{2}\sin \theta$ ): $\Big(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\Big)^{3}+\dfrac {z^{6}}{c^{6}}=\dfrac {xyz}{h^{3}}$ .

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Problem 3705

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}\Big)^{2}=\dfrac {z}{l}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3706

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}\Big)^{2}=\dfrac {x}{h}+\dfrac {y}{k}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3707

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}\Big)^{2}=\dfrac {x}{h}-\dfrac {y}{k}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3708

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}\Big)^{2}=\dfrac {x}{a}+\dfrac {y}{b}-\dfrac {z}{h}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3709

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}\Big)^{6}=\dfrac {xyz}{h^{3}}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3710

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}\Big)^{2}+\dfrac {z^{2}}{c^{2}}=\dfrac {x}{h}+\dfrac {y}{k}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3711

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}\Big)^{2}+\dfrac {z^{2}}{c^{2}}=\dfrac {x}{h}-\dfrac {y}{k}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3712

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}\Big)^{3}=\ln \dfrac {\frac{x}{a}+\frac{y}{b}+\frac{z}{c}}{\frac{x}{a}+\frac{y}{b}}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3713

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}\Big)^{3}=\sin \dfrac {\pi \big(\frac{x}{a}+\frac{y}{b}\big)}{\frac{x}{a}+\frac{y}{b}+\frac{z}{c}}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3714

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\Big(\dfrac {x}{a}\Big)^{2/3}+\Big(\dfrac {y}{b}\Big)^{2/3}+\Big(\dfrac {z}{c}\Big)^{2/3}=1$ .

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Problem 3715

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}+\sqrt{\dfrac {z}{c}}=1$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3716

Find the volume of the solid bounded by the given surface, introducing generalized spherical coordinates with exponent $\sigma$ by the formulas $x=ar\sin^{\sigma }\theta \cos^{\sigma }\varphi$ , $y=br\sin^{\sigma }\theta \sin^{\sigma }\varphi$ , $z=cr\cos^{\sigma }\theta$ (the Jacobian of this transformation is $J=abc\sigma^{2}r^{2}\sin^{2\sigma -1}\theta (\cos \theta \sin \varphi \cos \varphi )^{\sigma -1}$ ): $\sqrt[3]{\dfrac {x}{a}}+\sqrt[3]{\dfrac {y}{b}}+\sqrt[3]{\dfrac {z}{c}}=1$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3717

Using a change of variables, find the volume of the solid bounded by the surfaces: $x+y+z=a$ , $x+y+z=2a$ , $x+y=z$ , $x+y=2z$ , $y=x$ , $y=2x$ .

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Problem 3718

Using a change of variables, find the volume of the solid bounded by the surfaces: $a_{1}x+b_{1}y+c_{1}z=\pm h_{1}$ , $a_{2}x+b_{2}y+c_{2}z=\pm h_{2}$ , $a_{3}x+b_{3}y+c_{3}z=\pm h_{3}$ .

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Problem 3719

Using a change of variables, find the volume of the solid bounded by the surface $(a_{1}x+b_{1}y+c_{1}z)^{2}+(a_{2}x+b_{2}y+c_{2}z)^{2}+(a_{3}x+b_{3}y+c_{3}z)^{2}=1$ .

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Remark.

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Here $\Delta$ is the same determinant as in the previous problem: $\Delta =\begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}$ .

Problem 3720

Using a change of variables, find the volume of the solid bounded by the surfaces: $(a_{1}x+b_{1}y+c_{1}z)^{2}+(a_{2}x+b_{2}y+c_{2}z)^{2}=1$ , $a_{3}x+b_{3}y+c_{3}z=\pm h$ .

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Remark.

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Here $\Delta$ is the same determinant as in problem 3718: $\Delta =\begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}$ .

Problem 3721

Using a change of variables, find the volume of the solid bounded by the surface $\left|a_{1}x+b_{1}y+c_{1}z\right|+\left|a_{2}x+b_{2}y+c_{2}z\right|+\left|a_{3}x+b_{3}y+c_{3}z\right|=1$ .

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Remark.

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Here $\Delta$ is the same determinant as in problem 3718: $\Delta =\begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}$ .

Section 9.10Coordinates of Centers of Gravity and Moments of Inertia of Solids[37/86%]

Problem 3722

The coordinates $x_{c}$ , $y_{c}$ , $z_{c}$ of the center of gravity of a body occupying a volume $V$ are determined by the formulas:

x_{c}=\frac{1}{m}\iiint \limits _{V} \rho x \, dx \, dy \, dz, \qquad y_{c}=\frac{1}{m}\iiint \limits _{V} \rho y \, dx \, dy \, dz, \qquad z_{c}=\frac{1}{m}\iiint \limits _{V} \rho z \, dx \, dy \, dz,

where $\rho$ is the density of the body at the point $(x,y,z)$ , and $m$ is the mass of the body. In the following problems the body is assumed to be homogeneous with $\rho =1$ .

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x+y+z=a$ , $x=0$ , $y=0$ , $z=0$ .

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Problem 3723

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $z^{2}=xy$ , $x=a$ , $y=b$ , $z=c$ .

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Problem 3724

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $abz=c(a-x)(b-y)$ , $x=0$ , $y=0$ , $z=0$ .

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Problem 3725

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $h^{2}(x^{2}+y^{2})=a^{2}z^{2}$ ; $0<z<h$ .

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Problem 3726

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $x^{2}+y^{2}+z^{2}=a^{2}$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3727

Find the coordinates of the center of gravity of a homogeneous body that is a segment of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ ; $h<z<a$ .

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Problem 3728

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}+z^{2}=a^{2}$ , $x^{2}+y^{2}=ax$ .

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Problem 3729

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}=z$ , $x+y+z=0$ .

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Problem 3730

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}+z^{2}=a^{2}$ , $x^{2}+y^{2}=a(a-2z)$ .

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Problem 3731

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $x^{2}+y^{2}=2a^{2}$ , $x+y+z=2a$ , $x=0$ , $y=0$ , $z=0$ .

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Problem 3732

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $(x^{2}+y^{2}+z^{2})^{2}=axyz$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3733

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{3}z$ .

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Problem 3734

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z^{2}}{c^{2}}$ ; $z>0$ .

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Problem 3735

Find the coordinates of the center of gravity of a homogeneous body bounded by the surfaces $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z}{c}$ , $\dfrac {x}{a}+\dfrac {y}{b}=\pm 1$ , $\dfrac {x}{a}-\dfrac {y}{b}=\pm 1$ , $z=0$ .

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Problem 3736

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $\sqrt{\dfrac {x}{a}}+\sqrt{\dfrac {y}{b}}+\sqrt{\dfrac {z}{c}}=1$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3737

Find the coordinates of the center of gravity of a homogeneous body bounded by the surface $\Big(\dfrac {x}{a}\Big)^{2/3}+\Big(\dfrac {y}{b}\Big)^{2/3}+\Big(\dfrac {z}{c}\Big)^{2/3}=1$ ; $x>0$ , $y>0$ , $z>0$ .

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Problem 3738

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , where $l=\sqrt{x^{2}+y^{2}}$ is the distance from the variable point of the body to the axis $Oz$ , bounded by the surfaces $x=0$ , $x=a$ , $y=0$ , $y=b$ , $z=0$ , $z=c$ .

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Problem 3739

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , bounded by the surfaces $z^{2}=2ax$ , $z=0$ , $x^{2}+y^{2}=ax$ .

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Problem 3740

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , bounded by the surface $h^{2}(x^{2}+y^{2})=a^{2}z^{2}$ ; $0<z<h$ .

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Problem 3741

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , bounded by the surfaces $x+y+z=a\sqrt{2}$ , $x^{2}+y^{2}=a^{2}$ , $z=0$ .

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Problem 3742

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , bounded by the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{3}z$ .

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Problem 3743

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , bounded by the surface $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ .

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Problem 3744

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , bounded by the surfaces $\dfrac {x}{a}+\dfrac {y}{b}+\dfrac {z}{c}=1$ , $x=0$ , $y=0$ , $z=0$ .

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Problem 3745

Find the moment of inertia relative to the axis $Oz$ of a homogeneous body ( $\rho =1$ ), i.e., the integral $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , bounded by the surface $\Big(\dfrac {x}{a}\Big)^{2/3}+\Big(\dfrac {y}{b}\Big)^{2/3}+\Big(\dfrac {z}{c}\Big)^{2/3}=1$ .

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Problem 3746

Find the moments of inertia of the homogeneous torus

x=(a+r\cos \theta )\cos \varphi , \qquad y=(a+r\cos \theta )\sin \varphi , \qquad z=r\sin \varphi

relative to its axis of rotation.

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Problem 3747

Find the moment of inertia of the same homogeneous torus

x=(a+r\cos \theta )\cos \varphi , \qquad y=(a+r\cos \theta )\sin \varphi , \qquad z=r\sin \varphi ,

as in the previous problem, relative to its equatorial diameter.

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Problem 3748

Find the moment of inertia of the homogeneous elliptic cone $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z^{2}}{h^{2}}$ , $z=h$ , relative to the axis $Ox$ .

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Problem 3749

Find the moment of inertia of a homogeneous circular cylinder of height $H$ and base radius $a$ relative to a diameter of the base.

?

Problem 3750

Find the moment of inertia of a homogeneous circular cone of height $H$ and base radius $R$ relative to a diameter of the base.

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Problem 3751

(a)(b)

The moments of inertia of a homogeneous body ( $\rho =1$ ) relative to the coordinate axes are given by the formulas $I_{x}=\iiint \limits_{V}(y^{2}+z^{2}) \, dx \, dy \, dz$ , $I_{y}=\iiint \limits_{V}(z^{2}+x^{2}) \, dx \, dy \, dz$ , $I_{z}=\iiint \limits_{V}(x^{2}+y^{2}) \, dx \, dy \, dz$ , and the centrifugal moments of inertia by the integrals $I_{xy}=\iiint \limits_{V}xy \, dx \, dy \, dz$ , $I_{yz}=\iiint \limits_{V}yz \, dx \, dy \, dz$ , $I_{zx}=\iiint \limits_{V}zx \, dx \, dy \, dz$ .

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(a)

Prove that the moment of inertia of the body $J_{\sigma }$ relative to the axis $\dfrac {x}{\cos \alpha }=\dfrac {y}{\cos \beta }=\dfrac {z}{\cos \gamma }$ is expressed by the formula

J_{\sigma }=A\cos ^{2}\alpha +B\cos ^{2}\beta +C\cos ^{2}\gamma -2D\cos \alpha \cos \beta -2E\cos \alpha \cos \gamma -2F\cos \beta \cos \gamma ,

where the coefficients equal the corresponding moments of inertia and centrifugal moments: $A=I_{x}$ , $B=I_{y}$ , $C=I_{z}$ , $D=I_{xy}$ , $E=I_{xz}$ , $F=I_{yz}$ .

(b)

On each line $\sigma$ passing through the center of gravity of the body, a segment of length $\dfrac {1}{\sqrt{J_{\sigma }}}$ is laid off. Prove that the endpoints of these segments lie on the ellipsoid

Ax^{2}+By^{2}+Cz^{2}-2Dxy-2Exz-2Fyz=1;

this ellipsoid is called the ellipsoid of inertia of the given body.

Problem 3752

Determine the height $h$ and base radius $a$ of a homogeneous cylinder so that the ellipsoid of inertia for this cylinder becomes a sphere.

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Problem 3753

The same question as in the previous problem (determine the height $h$ and base radius so that the ellipsoid of inertia becomes a sphere), for a homogeneous cone.

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Problem 3754

Determine the height $h$ of a rectangular parallelepiped, whose base is a square with side $a$ , so that the ellipsoid of inertia for the homogeneous mass filling this parallelepiped becomes a sphere.

?

Problem 3755

Prove that among all homogeneous ellipsoids, only for the sphere is the ellipsoid of inertia itself a sphere.

?

Problem 3756

Prove that for homogeneous bodies symmetric with respect to the plane $yOz$ , the quantity $J_{xy}=0$ , where $J_{xy}$ is the centrifugal moment of inertia $I_{xy}=\iiint \limits_{V}xy \, dx \, dy \, dz$ (notation from problem 3751).

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Problem 3757

Prove that for homogeneous regular polyhedra, the ellipsoid of inertia is a sphere.

?

Problem 3758

Prove that a homogeneous ellipsoid can be similar to its own ellipsoid of inertia only in the case where the ellipsoid is a sphere.

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Section 9.11Integrals of Field Theory and Potential Theory[57/39%]

Problem 3759

If $u(x,y,z)$ is a function of a point, then the vector whose projections on the axes $Ox$ , $Oy$ , and $Oz$ equal the quantities $\dfrac {\partial u}{\partial x}$ , $\dfrac {\partial u}{\partial y}$ , $\dfrac {\partial u}{\partial z}$ is called the gradient of $u$ and denoted $\operatorname {grad}u$ .

Verify that $\operatorname {grad}(x^{2}+y^{2}+z^{2})=2\overrightarrow {r}$ , where $\overrightarrow {r}$ is the radius vector of the point $(x,y,z)$ .

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Problem 3760

Verify that $\operatorname {grad}\dfrac {1}{\sqrt{x^{2}+y^{2}+z^{2}}}=\dfrac {\overrightarrow {r}}{r^{3}}$ , where $\overrightarrow {r}$ is the radius vector of the point $(x,y,z)$ , and $r=\left|\overrightarrow {r}\right|$ .

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Problem 3761

Verify that $\operatorname {div}\operatorname {grad}(x^{2}+y^{2}+z^{2})=6$ , where for a vector $\overrightarrow {u}$ with projections $u_{x}$ , $u_{y}$ , $u_{z}$ , the divergence of the field is called the scalar quantity $\operatorname {div}\overrightarrow {u}=\dfrac {\partial u_{x}}{\partial x}+\dfrac {\partial u_{y}}{\partial y}+\dfrac {\partial u_{z}}{\partial z}$ .

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Problem 3762

Verify that $\operatorname {div}\dfrac {\overrightarrow {r}}{r^{3}}=0$ , where $\overrightarrow {r}$ is the radius vector of the point $(x,y,z)$ , and $r=\left|\overrightarrow {r}\right|$ .

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Problem 3763

Prove that if $u_{x}=-\dfrac {ay}{x^{2}+y^{2}}$ , $u_{y}=\dfrac {ax}{x^{2}+y^{2}}$ , $u_{z}=0$ , then the circulation of $\overrightarrow {u}$ along a closed curve (i.e., the line integral $\int_{L} \overrightarrow {u}\cdot d\overrightarrow {r}=\int_{L} u_{x} \, dx+u_{y} \, dy+u_{z} \, dz$ ) equals zero if the curve does not go around the axis $Oz$ . It equals $2\pi n$ if the curve goes around the axis $Oz$ $n$ times, in such a way that its projection onto the plane $xOy$ goes around the origin counterclockwise $n$ times.

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Problem 3764

Prove that for the vector $\overrightarrow {u}$ of the previous problem ( $u_{x}=-\dfrac {ay}{x^{2}+y^{2}}$ , $u_{y}=\dfrac {ax}{x^{2}+y^{2}}$ , $u_{z}=0$ ), the curl $\operatorname {rot}\overrightarrow {u}=0$ at all points except those on the axis $Oz$ , where for a vector $\overrightarrow {u}$ with projections $u_{x}$ , $u_{y}$ , $u_{z}$ , the curl (rotation) of the field is called the vector with projections on the axes $Ox$ , $Oy$ , $Oz$ equal to the quantities $\dfrac {\partial u_{z}}{\partial y}-\dfrac {\partial u_{y}}{\partial z}$ , $\dfrac {\partial u_{x}}{\partial z}-\dfrac {\partial u_{z}}{\partial x}$ , $\dfrac {\partial u_{y}}{\partial x}-\dfrac {\partial u_{x}}{\partial y}$ .

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Problem 3765

A medium performs a rotation, as a rigid body, about the axis $Oz$ with angular velocity $\omega$ . The velocity vector $\overrightarrow {v}$ has projections on the axes $v_{x}=-\omega y$ , $v_{y}=\omega x$ , $v_{z}=0$ . Find $\operatorname {rot}\overrightarrow {v}$ .

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Problem 3766

Find the circulation of the vector $\overrightarrow {v}$ of the previous problem ( $v_{x}=-\omega y$ , $v_{y}=\omega x$ , $v_{z}=0$ , the velocity vector of a medium rotating as a rigid body about the axis $Oz$ with angular velocity $\omega$ ) along the circle $x^{2}+y^{2}=a^{2}$ in the positive direction.

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Problem 3767

Find the circulation of the same vector $\overrightarrow {v}$ ( $v_{x}=-\omega y$ , $v_{y}=\omega x$ , $v_{z}=0$ , the velocity vector of a medium rotating as a rigid body about the axis $Oz$ with angular velocity $\omega$ ) along the circle $(x-2)^{2}+y^{2}=2$ .

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Problem 3768

The formula of Ostrogradsky-Gauss relates the triple integral over a volume $V$ to the double integral over the surface $\sigma$ bounding it:

\iiint \limits _{V}\Big(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\Big) \, dv = \iint \limits _{\sigma }\big(P\cos (x,\overrightarrow {n})+Q\cos (y,\overrightarrow {n})+R\cos (z,\overrightarrow {n})\big) \, d\sigma ,

where $\overrightarrow {n}$ is the unit outward normal vector to $\sigma$ , and $P$ , $Q$ , $R$ are the projections of the field vector $\overrightarrow {u}$ on the axes $Ox$ , $Oy$ , $Oz$ ; in vector form the same formula reads $\iiint_{V}\operatorname {div}\overrightarrow {u} \, dv=\iint_{\sigma }\overrightarrow {u}\cdot \overrightarrow {n} \, d\sigma =\iint_{\sigma }u_{n} \, d\sigma$ .

Prove the equality $\iint_{S}\cos (\overrightarrow {n},x) \, d\sigma =0$ , where $S$ is a closed surface and $\overrightarrow {n}$ is the outward normal to it.

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Problem 3769

Find the value of the integral $\displaystyle \iint \limits_{S}(x\cos \alpha +y\cos \beta +z\cos \gamma ) \, ds$ , where $\alpha$ , $\beta$ , $\gamma$ are the angles of the outward normal to the surface $S$ , and the integral is taken over the entire surface $S$ of some body of volume $v$ .

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Problem 3770

Find the integral $\displaystyle \iint \limits_{S}(x^{3}\cos \alpha +y^{3}\cos \beta +z^{3}\cos \gamma ) \, ds$ , taken over the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ . Here $\alpha$ , $\beta$ , $\gamma$ are the angles of the outward normal with the axes.

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Problem 3771

Find the integral $\displaystyle \iint \limits_{S}\big((z^{n}-y^{n})\cos \alpha +(x^{n}-z^{n})\cos \beta +(y^{n}-x^{n})\cos \gamma \big) \, ds$ , taken over the upper half of the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ ; $\alpha$ , $\beta$ , $\gamma$ are the angles of the normal outward from the sphere.

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Problem 3772

Applying the formula of Ostrogradsky-Gauss

\iiint \limits _{V}\Big(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\Big) \, dv = \iint \limits _{\sigma }\big(P\cos (x,\overrightarrow {n})+Q\cos (y,\overrightarrow {n})+R\cos (z,\overrightarrow {n})\big) \, d\sigma

to the vector $u\operatorname {grad}v$ , obtain Green’s formula:

\iiint \limits _{\omega }\Big(u\Delta v+\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}+\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}\Big) \, d\omega = -\iint \limits _{\sigma }u\frac{\partial u}{\partial n} \, d\sigma .

Here $\omega$ is a volume; $\sigma$ is its surface; $\dfrac {\partial v}{\partial n}=\dfrac {\partial v}{\partial x}\cos \alpha +\dfrac {\partial v}{\partial y}\cos \beta +\dfrac {\partial v}{\partial z}\cos \gamma$ is the derivative along the inward normal, and $\Delta v=\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}+\dfrac {\partial^{2}v}{\partial z^{2}}$ .

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Problem 3773

Starting from the previous result (Green's formula, problem 3772: $\iiint_{\omega }\big(u\Delta v+\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}+\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}\big) \, d\omega = -\iint_{\sigma }u\frac{\partial u}{\partial n} \, d\sigma$ ), prove Green’s second formula:

\iiint \limits _{\omega }(u\Delta v-v\Delta u) \, d\omega = -\iint \limits _{\sigma }\Big(u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n}\Big) \, d\sigma .

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Problem 3774

A function $u$ for which $\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}+\dfrac {\partial^{2}v}{\partial z^{2}}=0$ is called harmonic. Prove that for harmonic functions the formula

\iiint \limits _{\omega }\left|\operatorname {grad}u\right|^{2} \, d\omega = -\iint \limits _{\sigma }u\frac{\partial v}{\partial n} \, d\sigma

holds.

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Problem 3775

Prove a formula similar to the formula of Ostrogradsky-Gauss

\iiint \limits _{V}\Big(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\Big) \, dv = \iint \limits _{\sigma }\big(P\cos (x,\overrightarrow {n})+Q\cos (y,\overrightarrow {n})+R\cos (z,\overrightarrow {n})\big) \, d\sigma :

\iint \limits _{\sigma }\Big(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}\Big) \, d\sigma = \int \limits _{S}(u\cos \alpha +v\cos \beta ) \, ds.

Here $\sigma$ is an area, and $S$ is the line bounding it; $\alpha$ and $\beta$ are the angles of the outward normal with the axes.

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Problem 3776

Prove a formula similar to Green's formula ( $\iiint_{\omega }\big(u\Delta v+\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}+\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}\big) \, d\omega = -\iint_{\sigma }u\frac{\partial u}{\partial n} \, d\sigma$ , problem 3772):

\iint \limits _{\sigma }\Big(u\Delta v+\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}\Big) \, d\sigma = \int \limits _{S}u\frac{\partial u}{\partial n} \, ds.

Here $\sigma$ is an area, $S$ is its contour, $n$ is the inward normal, $\Delta v=\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}$ .

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Problem 3777

Prove the equality:

\iint \limits _{\sigma }(u\Delta v-v\Delta u) \, d\sigma = -\int \limits _{S}\Big(u\frac{\partial v}{\partial n}+v\frac{\partial u}{\partial n}\Big) \, ds,

where $\sigma$ is an area, $S$ is its contour, $n$ is the inward normal, $\Delta v=\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}$ and similarly for $\Delta u$ (notation of problem 3776).

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Problem 3778

Prove that for harmonic functions of two variables (i.e., functions $u$ for which $\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}=0$ ) the equality

\iint \limits _{\sigma }\bigg(\Big(\frac{\partial u}{\partial x}\Big)^{2}+\Big(\frac{\partial u}{\partial y}\Big)^{2}\bigg) \, d\sigma = -\int \limits _{S}u\frac{\partial u}{\partial n} \, ds

holds.

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Problem 3779

For the same functions as in the previous problem (harmonic functions of two variables), prove the formula

\int \limits _{S}\frac{\partial u}{\partial n} \, ds = 0.

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Problem 3780

Prove that for harmonic functions of three variables

\iint \limits _{\sigma }\frac{\partial u}{\partial n} \, d\sigma = 0.

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Problem 3781

Using Green's second formula ( $\iiint_{\omega }(u\Delta v-v\Delta u) \, d\omega = -\iint_{\sigma }\big(u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n}\big) \, d\sigma$ , see 3773), applied to the region between the surface $\sigma$ and a sphere of radius $\rho \to 0$ centered at the point $M(\xi ,\eta ,\zeta )$ lying inside the surface $\sigma$ , prove for a harmonic function of three variables the equality

u(\xi ,\eta ,\zeta ) = \frac{1}{4\pi }\iint \limits _{\sigma } u\frac{\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}} \, d\sigma + \frac{1}{4\pi }\iint \limits _{\sigma }\frac{\partial u}{\partial n}\frac{d\sigma }{r},

where $r$ is the distance from the point $M$ to a variable point of the surface, and $\overrightarrow {n}$ is the outward normal vector.

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Problem 3782

Prove that

\iint \limits _{\sigma }\frac{\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}} \, d\sigma = 4\pi \text{ or } 0,

according to whether the point from which the radius vectors $\overrightarrow {r}$ are drawn to the points of the surface lies inside the surface or outside it. The surface is assumed to be simply connected, closed, and without singular points.

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Problem 3783

Prove that for harmonic functions of two variables the formula

u(x,y) = \frac{1}{2\pi }\int \limits _{S}\Big(\ln r\frac{\partial u}{\partial n}-u\frac{\partial \ln r}{\partial n}\Big) \, ds

holds, where the derivatives are taken along the inward normal, and the point $M(x,y)$ is inside the contour $S$ .

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Problem 3784

Prove the equality

\int \limits _{AB}\frac{\cos (\overrightarrow {r},\overrightarrow {n})}{r} \, ds = \varphi ,

where $\varphi$ is the angle at which the line $AB$ is seen from the point $(\xi ,\eta )$ , $r=\sqrt{(x-\xi )^{2}+(y-\eta )^{2}}$ , and $\overrightarrow {n}$ is the normal to the line $AB$ .

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Problem 3785

Prove that for harmonic functions of three variables the formula

u(x,y,z) = \frac{1}{4\pi r^{2}}\iint \limits _{\sigma } u \, d\sigma

holds, where $\sigma$ is the surface of a sphere of radius $r$ centered at the point $(x,y,z)$ . (Gauss.)

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Problem 3786

Prove for harmonic functions of two variables the formula

u(x,y) = \frac{1}{2\pi r^{2}}\int \limits _{S} u \, ds,

where $S$ is the circle of radius $r$ centered at the point $(x,y)$ .

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Problem 3787

Prove that for a function $u(x,y,z)$ harmonic inside a sphere of volume $\omega$ , the formula

u_{c} = \frac{1}{\omega }\iiint \limits _{\omega } u \, d\omega

holds, where $u_{c}$ is the value of $u$ at the center of the sphere; and for a function $u(x,y)$ harmonic inside a disk of area $\sigma$ , prove the formula

u_{c} = \frac{1}{\sigma }\iint \limits _{\sigma } u \, d\sigma ,

where $u_{c}$ is the value of $u$ at the center of the disk.

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Problem 3788

Using the Taylor series expansion, prove the equality

\frac{1}{2\pi r}\int \limits _{S} u \, du = u + \sum _{n=1}^{\infty }\frac{1}{(n)!}\Big(\frac{r}{2}\Big)^{2n}\Delta ^{n}u,

where

\Delta u = \frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial y^{2}}, \qquad \Delta ^{2}u = \Delta (\Delta u) = \frac{\partial ^{4}u}{\partial x^{4}}+2\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{\partial ^{4}u}{\partial y^{4}}

and so on, and the integration is carried out over the circle of radius $r$ centered at the point $(x,y)$ .

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Problem 3789

Prove the equality

\frac{1}{4\pi r^{2}}\iint \limits _{\sigma } u \, d\sigma = u(x,y,z) + \sum _{n=1}^{\infty }\frac{r^{2n}}{(2n+1)!}\Delta ^{n}u,

where

\Delta u = \frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\partial ^{2}u}{\partial z^{2}}, \qquad \Delta ^{2}u = \Delta (\Delta u), \qquad \Delta ^{3}u = \Delta (\Delta ^{2}u), \ldots ,

and the integration is taken over the surface of a sphere of radius $r$ centered at the point $(x,y,z)$ .

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Problem 3790

Prove Maxwell's formula

\iiint \limits _{\omega }\frac{\partial u}{\partial x}\Delta u \, d\omega = \frac{1}{2}\iint \limits _{\sigma }\bigg(\Big(\frac{\partial u}{\partial x}\Big)^{2}-\Big(\frac{\partial u}{\partial y}\Big)^{2}-\Big(\frac{\partial u}{\partial z}\Big)^{2}\bigg)\cos (\overrightarrow {n},x) \, d\sigma +{}

{}+\iint \limits _{\sigma }\frac{\partial u}{\partial x}\frac{\partial u}{\partial y}\cos (\overrightarrow {n},y) \, d\sigma + \iint \limits _{\sigma }\frac{\partial u}{\partial x}\frac{\partial u}{\partial z}\cos (\overrightarrow {n},z) \, d\sigma .

Here $\sigma$ is the surface of the volume $\omega$ , in which the function $u$ is continuous together with its second derivatives.

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Problem 3791

Prove Riemann's equality

\iint \limits _{\sigma }\big(uF(v)-vG(u)\big) \, d\sigma = \int \limits _{S} M \, dx+N \, dy.

Here $S$ is the contour of the area $\sigma$ ,

F(v) = \frac{\partial ^{2}v}{\partial x \partial y}-a\frac{\partial v}{\partial x}-b\frac{\partial v}{\partial y}-cv, \qquad G(u) = \frac{\partial ^{2}u}{\partial x \partial y}+a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}-cu,

and the expressions for $M$ and $N$ are to be found.

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Problem 3792

If the projections of the velocity of a fluid at time $t$ at the point $(x,y,z)$ equal the functions $u(x,y,z,t)$ , $v(x,y,z,t)$ , and $w(x,y,z,t)$ , then the amount of fluid flowing through the surface $\sigma$ during the time from $t_{1}$ to $t_{2}$ equals the integral

\int \limits _{t_{1}}^{t_{2}} dt \iint \limits _{\sigma }(u\cos \alpha +v\cos \beta +w\cos \gamma ) \, d\sigma ,

where $\alpha$ , $\beta$ , $\gamma$ are the angles of the normal to the surface $\sigma$ with the axes. Deduce from this that for an incompressible fluid the condition

\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} = 0

must hold.

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Problem 3793

Study the variation of the function $P(x)$ , where $P(x)=\mu \int_{a}^{b}\left|y-x\right| \, dy$ , as $x$ varies over the interval $(-\infty ,+\infty )$ .

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Problem 3794

Find the second derivative of

P(x) = \int \limits _{a}^{b} \mu (y)\left|y-x\right| \, dy,

where the function $\mu (y)$ is continuous.

?

Problem 3795

Using Poisson's integral

\int \limits _{0}^{\pi }\ln (a^{2}-2a\cos \varphi +1) \, d\varphi = \begin{cases} 0, & \text{if } -1<a<1; \\ 2\pi \ln \left|a\right|, & \text{if } \left|a\right|>1, \end{cases}

find the logarithmic potential of a homogeneous disk, i.e., the integral $\mu \iint_{\sigma } \ln r \, d\sigma$ , where $r$ is the distance between a point $M$ and a point on the area of a disk $\sigma$ of radius $R$ .

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Problem 3796

The logarithmic potential of a homogeneous segment $(-1,1)$ is called the integral $P=\int_{-1}^{+1}\ln r \, d\xi$ , where $r$ is the distance between the points $(x,y)$ and $(\xi ,0)$ . Find the equation of the curve on which $P(x,y)=P(1,0)$ .

?

Problem 3797

The logarithmic potential of an area $\sigma$ at the point $(x,y)$ is called the integral

P(x,y) = \iint \limits _{\sigma } \mu (\xi ,\eta )\ln \sqrt{(x-\xi )^{2}+(y-\eta )^{2}} \, d\xi \, d\eta .

Find the logarithmic potential of the disk $x^{2}+y^{2}=a^{2}$ , if the density $\mu$ at the point $(x,y)$ is a given function of the radius vector from the origin: $\mu =f(r)$ , where $r=\sqrt{x^{2}+y^{2}}$ .

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Problem 3798

Find at the point $M(x,y,z)$ the Newtonian potential of a homogeneous shell of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ with surface density $\mu$ equal to some constant, i.e., find the integral $\iint_{\sigma }\mu \dfrac {d\sigma }{r}$ , where $\sigma$ is the surface of the sphere, and $r$ is the distance from a variable point $(\xi ,\eta ,\zeta )$ on this surface to the given point $M$ .

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Problem 3799

Find at the point $M(x,y,z)$ the Newtonian potential of a homogeneous sphere, i.e., the integral

P = \iiint \limits _{\omega } \mu \frac{d\xi \, d\eta \, d\zeta }{r},

where $\omega$ is a sphere of radius $a$ centered at the origin, $\mu$ is the constant density, $(\xi ,\eta ,\zeta )$ is a point inside the sphere, $r=\sqrt{(x-\xi )^{2}+(y-\eta )^{2}+(z-\zeta )^{2}}$ is the distance from the point $(\xi ,\eta ,\zeta )$ to the point $(x,y,z)$ .

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Problem 3800

The same question (find at the point $M(x,y,z)$ the Newtonian potential of a sphere, i.e., the integral $P=\iiint_{\omega } \mu \, d\xi \, d\eta \, d\zeta / r$ , where $\omega$ is a sphere of radius $a$ centered at the origin, $r$ is the distance from the point $(\xi ,\eta ,\zeta )$ inside the sphere to the point $(x,y,z)$ ) for a sphere whose density $\mu$ equals a given function $f(\rho )$ , where $\rho^{2}=\xi^{2}+\eta^{2}+\zeta^{2}$ .

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Problem 3801

At the point $(0,0,z)$ find the Newtonian potential of the upper half of the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ with constant density $\mu$ .

?

Problem 3802

Find the Newtonian potential of a homogeneous spherical shell between two concentric spheres of radii $a$ and $b$ , where $a<b$ .

?

Problem 3803

Find at the apex of a cone the Newtonian potential of the solid cone with base radius $r$ and height $h$ .

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Problem 3804

Show that among bodies of constant density and given mass, the one that attracts a material point most strongly (by Newton's law) is bounded by the surface $R=a\sqrt{\cos \theta }$ (the material point is taken as the origin, the direction of the attractive force as the axis $Oz$ ; $R$ , $\theta$ are spherical coordinates). Find this force and compare it with the force of attraction by a sphere (of the same mass) on a material point lying on its surface.

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Problem 3805

Find at the point $M(x,y)$ the Newtonian potential of a non-homogeneous segment $(-a,a)$ , i.e., the integral $\displaystyle \int_{-a}^{a}\frac{\mu (\xi ) \, d\xi }{\sqrt{(\xi -x)^{2}+y^{2}}}$ , if $\mu (\xi )=\dfrac {\xi }{a}$ .

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Problem 3806

Find the Newtonian potential of a homogeneous circular plate $x^{2}+y^{2}=a^{2}$ , $z=0$ , at the point $(0,0,z)$ .

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Problem 3807

Prove the equality

\lim _{r_{1}\to \infty } r_{1}\iiint \limits _{\omega } \mu \frac{d\omega }{r} = \iiint \limits _{\omega } \mu \, d\omega .

Here $r_{1}^{2}=x^{2}+y^{2}+z^{2}$ , $r$ is the distance from the point $(x,y,z)$ to an arbitrary point $(\xi ,\eta ,\zeta )$ inside the volume $\omega$ .

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Problem 3808

Prove the identity

\iiint \limits _{\omega } \frac{d\xi \, d\eta \, d\zeta }{r} = \frac{1}{2}\iint \limits _{\sigma } \cos (\overrightarrow {r},\overrightarrow {n}) \, d\sigma ,

where $\sigma$ is the surface of the body of volume $\omega$ ; $\overrightarrow {n}$ is the outward normal to $\sigma$ ; $\overrightarrow {r}$ is the vector from the point $(x,y,z)$ to the point $(\xi ,\eta ,\zeta )$ ; $r$ is its length.

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Problem 3809

Under the same conditions as in the previous problem ( $\sigma$ is the surface of the body of volume $\omega$ ; $\overrightarrow {n}$ is the outward normal to $\sigma$ ; $\overrightarrow {r}$ is the vector from the point $(x,y,z)$ to the point $(\xi ,\eta ,\zeta )$ ; $r$ is its length), prove the identity

\iiint \limits _{\omega } \frac{d\xi \, d\eta \, d\zeta }{r^{2}} = \iint \limits _{\sigma } \frac{\cos (\overrightarrow {r},\overrightarrow {n})}{r} \, d\sigma .

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Problem 3810

Write down analogous identities (see problem 3809) for an area $\sigma$ and its contour $S$ .

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Problem 3811

Prove the equality $\displaystyle \int_{S} \frac{\cos (\overrightarrow {r},\overrightarrow {n})}{r} \, ds = 0$ , if the point $(x,y)$ lies outside the region bounded by the contour $S$ , and $=2\pi$ , if the contour $S$ makes one complete circuit around the point $(x,y)$ .

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Problem 3812

The projections on the axes $Ox$ , $Oy$ , $Oz$ of the force with which a point of mass $m$ is attracted, by Newton's law, by a body of density $\mu$ , are expressed by the formulas:

X = fm\frac{\partial P}{\partial x} = fm\iiint \limits _{\omega } \mu \frac{(\xi -x) \, d\omega }{r^{3}}, \qquad Y = fm\frac{\partial P}{\partial y} = fm\iiint \limits _{\omega } \mu \frac{(\eta -y) \, d\omega }{r^{3}},

Z = fm\frac{\partial P}{\partial z} = fm\iiint \limits _{\omega } \mu \frac{(\zeta -z) \, d\omega }{r^{3}},

where $P=\iiint_{\omega } \mu \dfrac {d\omega }{r}$ , $r=\sqrt{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}}$ , $f$ is the gravitational constant. Using this, compute the force of attraction by a sphere of radius $a$ with density $\mu$ on a point of mass $m$ located on its surface.

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Problem 3813

By the Biot-Savart law, each element $d\overrightarrow {s}$ of a current of strength $I$ creates at the point $M(x,y,z)$ a magnetic field whose intensity can be expressed by the vector product $-\dfrac {kI}{r^{2}}d\overrightarrow {s}\times \overrightarrow {r}$ , where $\overrightarrow {r}$ is the vector going from $d\overrightarrow {s}$ to $M$ . Prove that the field intensity $\overrightarrow {H}$ created by a closed current flowing along the contour $S$ is expressed by the vector integral $\overrightarrow {H}=-kI\int_{S} \dfrac {d\overrightarrow {s}\times \overrightarrow {r}}{r^{2}}$ ; here $k$ is the proportionality coefficient.

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Problem 3814

Using the previous result (problem 3813: $\overrightarrow {H}=-kI\int_{S} \frac{d\overrightarrow {s}\times \overrightarrow {r}}{r^{2}}$ , the intensity of the magnetic field of a closed current by the Biot-Savart law) and Stokes's formula

\int \limits _{L} \overrightarrow {u}\cdot d\overrightarrow {r} = \iint \limits _{\sigma } \operatorname {rot}\overrightarrow {u}\cdot \overrightarrow {n} \, d\sigma = \iint \limits _{\sigma } \operatorname {rot}_{n}\overrightarrow {u} \, d\sigma

(the circulation of a vector around a closed contour $L$ equals the flux of the curl of the vector through the surface $\sigma$ bounded by the contour), prove the equality $\overrightarrow {H}=-kI\operatorname {grad}W$ , where $W$ is the solid angle at which the area $\sigma$ with contour $S$ is seen from the point $M$ , equal to the integral $\iint_{\sigma } \dfrac {\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}} \, d\sigma$ .

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Problem 3815

Consider the integral $\displaystyle \int_{s_{1}} d\overrightarrow {s}_{1}\int_{s_{2}} \frac{d\overrightarrow {s}_{2}\times \overrightarrow {r}}{r^{3}}$ , taken over two closed contours $s_{1}$ and $s_{2}$ , where $\overrightarrow {r}$ is the vector joining variable points on the contours. Regarding $\dfrac {d\overrightarrow {s}_{1}\times d\overrightarrow {s}_{2}\cdot \overrightarrow {e}_{r}}{r^{2}}$ as a solid angle, prove that this double integral equals $0$ if one contour is not linked with the other, and equals $\pm 4\pi$ if it is linked (once).

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Section 9.12Multiple (Iterated) Integrals[12/58%]

Problem 3816

Find the multiple integral $\displaystyle \int \cdots \int dx_{1} \, dx_{2}\ldots dx_{n}$ , taken over the region where all $x_{k}>0$ and $x_{1}+x_{2}+\ldots +x_{n}\leqslant 1$ .

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Problem 3817

Find the multiple integral $\displaystyle \int \cdots \int x_{1} \, dx_{1} \, dx_{2}\ldots dx_{n}$ , taken over the region where $x_{k}>0$ and $x_{1}+x_{2}+\ldots +x_{n}\leqslant 1$ .

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Problem 3818

Find the multiple integral

\int \limits _{\left|x_{1}\right|+\left|x_{2}\right|+\ldots +\left|x_{n}\right|<a}\cdots \int dx_{1} \, dx_{2}\ldots dx_{n}.

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Problem 3819

Find the multiple integral $\displaystyle \int_{0}^{1}\int_{0}^{1}\cdots \int_{0}^{1}(x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}) \, dx_{1} \, dx_{2}\ldots dx_{n}$ .

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Problem 3820

Find the multiple integral $\displaystyle \int_{0}^{1}\int_{0}^{1}\cdots \int_{0}^{1}(x_{1}x_{2}+x_{1}x_{3}+\ldots +x_{n-1}x_{n}) \, dx_{1} \, dx_{2}\ldots dx_{n}$ .

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Problem 3821

Let

u_{n}(a) = \int \limits _{x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}<a^{2}}\cdots \int dx_{1} \, dx_{2}\ldots dx_{n}, \qquad v_{n}=u_{n}(1).

Prove the equality $u_{n}(a)=a^{n}v_{n}$ .

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Problem 3822

Using the previous notation (problem 3821: $u_{n}(a)=\int \cdots \int dx_{1} \, dx_{2}\ldots dx_{n}$ , taken over the region $x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}<a^{2}$ , and $v_{n}=u_{n}(1)$ ), the following equalities are obvious:

v_{n} = \iint \limits _{x_{1}^{2}+x_{2}^{2}<1} dx_{1} \, dx_{2} \int \limits _{x_{3}^{2}+x_{4}^{2}+\ldots +x_{n}^{2}<1-x_{1}^{2}-x_{2}^{2}}\cdots \int dx_{3} \, dx_{4}\ldots dx_{n} ={}

{}= \iint \limits _{x_{1}^{2}+x_{2}^{2}<1} (1-x_{1}^{2}-x_{2}^{2})^{\frac{n}{2}-1}v_{n-2} \, dx_{1} \, dx_{2}.

Using these, prove that

u_{n}(a) = \frac{\pi ^{n/2}a^{n}}{\Gamma \big(\frac{n}{2}+1\big)}.

In particular, $u_{1}=2a$ , $u_{2}=\pi a^{2}$ , $u_{3}=\dfrac {4}{3}\pi a^{3}$ , $u_{4}=\dfrac {\pi^{2}}{2}a^{4}, \ldots$

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Problem 3823

If

u_{n} = \int \limits _{\substack {x_{1}+x_{2}+\ldots +x_{n}<a \\ 0<x_{1},0<x_{2},\ldots ,0<x_{n}}}\cdots \int x_{1}^{\alpha _{1}-1}x_{2}^{\alpha _{2}-1}\ldots x_{n}^{\alpha _{n}-1} \, dx_{1} \, dx_{2}\ldots dx_{n},

then it is easy to prove that $u_{n}(a)=a^{\alpha_{1}+\alpha_{2}+\ldots +\alpha_{n}}v_{n}$ , where $v_{n}=u_{n}(1)$ . Using this and the equality

v_{n} = \int \limits _{0}^{1} x_{1}^{\alpha _{1}-1} \, dx_{1} \int \limits _{\substack {x_{2}^{2}+x_{3}^{2}+\ldots +x_{n}^{2}<1-x_{1}^{2} \\ 0<x_{2},\ldots ,0<x_{n}}}\cdots \int x_{2}^{\alpha _{2}-1}x_{3}^{\alpha _{3}-1}\ldots x_{n}^{\alpha _{n}-1} \, dx_{2} \, dx_{3}\ldots dx_{n},

prove that

u_{n}(a) = \frac{\Gamma (\alpha _{1})\Gamma (\alpha _{2})\ldots \Gamma (\alpha _{n})}{\Gamma (1+\alpha _{1}+\alpha _{2}+\ldots +\alpha _{n})}a^{\alpha _{1}+\alpha _{2}+\ldots +\alpha _{n}}.

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Problem 3824

Prove the equality:

\int \limits _{0}^{x} dx_{1} \int \limits _{0}^{x_{1}} dx_{2}\ldots \int \limits _{0}^{x_{n-1}} f(x) \, dx_{n} = \int \limits _{0}^{x} f(t)\frac{(x-t)^{n-1}}{(n-1)!} \, dt.

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Problem 3825

Prove the equality:

\int \limits _{0}^{x} x_{1} \, dx_{1} \int \limits _{0}^{x_{1}} x_{2} \, dx_{2}\ldots \int \limits _{0}^{x_{n-1}} x_{n} \, dx_{n} \int \limits _{0}^{x_{n}} f(t) \, dt ={}

{}= \frac{1}{2\cdot 4\cdot \ldots \cdot 2n}\int \limits _{0}^{x} f(t)(x^{2}-t^{2})^{n} \, dt.

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Problem 3826

Reduce to a single integral the multiple integral

\iint \cdots \int x_{1}^{\alpha _{1}-1}x_{2}^{\alpha _{2}-1}\ldots x_{n}^{\alpha _{n}-1}F(x_{1}+x_{2}+\ldots +x_{n}) \, dx_{1} \, dx_{2}\ldots dx_{n},

taken over the region defined by the inequalities

0<x_{1}, \quad 0<x_{2}, \quad \ldots , \quad 0<x_{n}, \quad x_{1}+x_{2}+\ldots +x_{n}<a.

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Remark.

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One approach is a change of variables by the formulas $x_{1}+x_{2}+\ldots +x_{n}=u$ , $x_{1}=x_{1}$ , $x_{2}=x_{2}$ , $x_{3}=x_{3},\ldots ,x_{n-1}=x_{n-1}$ .

Problem 3827

The points $M(x,y,z)$ and $M_{1}(\xi ,\eta ,\zeta )$ independently of each other range over the entire volume of a body of density $p=p(x,y,z)$ . The self-potential of the body is called the integral

\frac{1}{2}\int \! \! \int \! \! \int \! \! \int \! \! \int \! \! \int \frac{p(x,y,z)p(\xi ,\eta ,\zeta ) \, dx \, dy \, dz \, d\xi \, d\eta \, d\zeta }{\sqrt{(x-\xi )^{2}+(y-\eta )^{2}+(z-\zeta )^{2}}}.

Find the value of the self-potential of a homogeneous sphere $x^{2}+y^{2}+z^{2}=a^{2}$ . Its value measures the work done in bringing from infinity into a single body particles that attract each other by Newton's law.

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Remark.

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Two approaches quickly lead to the goal: 1) Use the fact that the integral over the variables $(\xi ,\eta ,\zeta )$ gives the potential of the sphere at an interior point $(x,y,z)$ , found in one of the earlier problems (problem 3799). 2) Account for the work $dA$ done when the masses coming from infinity increase the radius of the sphere from $r$ to $r+dr$ , and integrate the result over $r$ from $0$ to $a$ .