Math Problems Vault

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 $x_{1}+x_{2}+\ldots +x_{n}\leqslant 1$ .

?

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 $x_{1}+x_{2}+\ldots +x_{n}\leqslant 1$ .

?

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}.

?

Problem 3819

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

?

Problem 3820

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

?

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

?

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

?

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}}.

?

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.

?

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.

?

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.

?

Remark.

?

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.

?

Remark.

?

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 $a$ .