Area, volume or arc length of a generalized ellipse via gamma and beta functions

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Problem 3879 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Find the arc length of the curve $r^{n}=a^{n}\cos n\varphi$ ( $a>0$ , $n$ a natural number).

Problem 3880 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Find the area enclosed by the curve $\left|x\right|^{n}+\left|y\right|^{n}=a^{n}$ ( $n>0$ , $a>0$ ).

Problem 4031 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.3 "Methods for Computing Volume"

Find the volume bounded by the following surfaces (parameters assumed positive): $z=x^{\frac{3}{2}}+y^{\frac{3}{2}}$ , $z=0$ , $x+y=1$ , $x=0$ , $y=0$ .

Problem 4034 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.3 "Methods for Computing Volume"

Find the volume bounded by the following surfaces (parameters assumed positive): $\dfrac {x^{n}}{a^{n}}+\dfrac {y^{n}}{b^{n}}+\dfrac {z^{n}}{c^{n}}=1$ , $x=0$ , $y=0$ , $z=0$ $(n>0)$ .

Problem 4035 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.3 "Methods for Computing Volume"

Find the volume bounded by the following surfaces (parameters assumed positive): $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{n}+\left(\dfrac {z}{c}\right)^{m}=1$ , $x=0$ , $y=0$ , $z=0$ $(n>0,m>0)$ .

Problem 4099 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.6 "Triple Integrals"

Find

\iiint \limits _{x^{2}+y^{2}+z^{2}\leqslant 1} x^{m}y^{n}z^{p}\, dx\, dy\, dz ,

where $m,n,p$ are nonnegative integers.

Problem 4100 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.6 "Triple Integrals"

Compute the Dirichlet integral

\iiint \limits _{V} x^{p}y^{q}z^{r}(1-x-y-z)^{s}\, dx\, dy\, dz \quad (p\geqslant 0, q\geqslant 0, r\geqslant 0, s\geqslant 0) ,

where $V$ is the region bounded by the planes $x+y+z=1$ , $x=0$ , $y=0$ , $z=0$ , given the substitutions

x+y+z=\xi , \quad y+z=\xi \eta , \quad z=\xi \eta \zeta .

Problem 4130 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.7 "Computing volumes by means of triple integrals"

Find the volume in the first octant $Oxyz$ $(x\geqslant 0, y\geqslant 0, z\geqslant 0)$ bounded by the surface

\dfrac {x^{m}}{a^{m}}+\dfrac {y^{n}}{b^{n}}+\dfrac {z^{p}}{c^{p}} = 1 \quad (m>0, n>0, p>0) , \quad x=0 , \quad y=0 , \quad z=0

Problem 4141 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.8 "Applications of Triple Integrals in Mechanics"

Find the coordinates of the centroid of the uniform solid bounded by the following surfaces: $\dfrac {x^{n}}{a^{n}}+\dfrac {y^{n}}{b^{n}}+\dfrac {z^{n}}{c^{n}}=1$ , $x=0$ , $y=0$ , $z=0$ $(n>0,\ x\geqslant 0,\ y\geqslant 0,\ z\geqslant 0)$ .

Problem 4211 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.10 "Multiple Integrals"

Find the volume of the $n$ -dimensional ball

x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leqslant a^{2} .

Problem 4216 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.10 "Multiple Integrals"

Prove the Dirichlet formula

\iint \limits _{\substack {x_{1},x_{2},\ldots ,x_{n}\geqslant 0 \\ x_{1}+x_{2}+\cdots +x_{n}\leqslant 1}}\cdots \int x_{1}^{p_{1}-1}x_{2}^{p_{2}-1}\cdots x_{n}^{p_{n}-1}\, dx_{1}\, dx_{2}\cdots dx_{n} = \dfrac {\Gamma (p_{1})\Gamma (p_{2})\cdots \Gamma (p_{n})}{\Gamma (p_{1}+p_{2}+\cdots +p_{n}+1)} \quad (p_{1},p_{2},\ldots ,p_{n}>0) .

Problem 4315 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 8.12 "Green's Formula"

Compute the area bounded by the quarter of the curve $\left(\dfrac {x}{a}\right)^{n}+\left(\dfrac {y}{b}\right)^{n}=1\ (a>0,\ b>0,\ n>0)$ lying in the first quadrant and the coordinate axes.

Problem 4759 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Find the area bounded by the curve $x^{n}+y^{n} = a^{n}$ , for $x > 0$ , $y > 0$ , $n > 0$ .

Problem 4760 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Find the volume bounded by the surface $x^{n}+y^{n}+z^{n} = a^{n}$ , for $x > 0$ , $y > 0$ , $z > 0$ , $n > 0$ .

Problem V.2.4 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section V.2 "Computing Multiple Integrals"

(a)(b)

Evaluate the integrals

(a)

$\iint_{\mathbb {R}^{2}}\left|a x+b y\right| \frac{d x d y}{e^{\left(x^{2}+y^{2}\right) / 2}}$ ;

(b)

$\int_{\mathbb {R}^{n}}\left|(a, x)\right|^{p} \frac{d x}{e^{\left\| x\right\|^{2} / 2}} \quad \left(a \in \mathbb {R}^{n}, p>-1\right)$ .

Problem VIII.3.25 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.3 "Summable Functions"

Compute the integral ( $p>-1$ )

\int _{\left[0,1\right]^{m}}\left(\frac{\min x_{k}}{\max x_{k}}\right)^{p} d x

Problem VIII.3.37 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.3 "Summable Functions"

(a)(b)

(a)

Compute the integral

\mathcal{Y}=\frac{1}{\sigma _{m-1}} \int _{S^{m-1}}\left|(x, a)\right|^{p} d \mu _{m-1}(x)

(b)

Prove that for $m, n \in \mathbb {N}, 1 \leqslant n \leqslant m, n+q>0$ , the following equality holds

\frac{1}{\sigma _{m-1}} \int _{S^{m-1}}\left(x_{1}^{2}+\ldots +x_{n}^{2}\right)^{q / 2} d \mu _{m-1}(x)=\frac{\Gamma \left(\frac{n+q}{2}\right)}{\Gamma \left(\frac{m+q}{2}\right)}-\frac{\Gamma \left(\frac{m}{2}\right)}{\Gamma \left(\frac{n}{2}\right)} \quad \left(\underset {n \rightarrow \infty }{\sim }\left(\frac{n}{m}\right)^{q / 2}\right)

Remark.

The symbol $\mu_{m-1}$ denotes the Lebesgue measure (surface area) on the sphere $S^{m-1}=\left\{ x \in \mathbb {R}^{m} \mid \left\| x\right\| =1\right\}$ .

Problem VIII.3.47 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.3 "Summable Functions"

(a)(b)(c)

Identifying $\mathbb {C}^{2}$ with $\mathbb {R}^{4}$ in the natural way, consider on the sphere $S^{3}$ the functions $\varphi_{m, n}(m, n=0,1,2, \ldots )$ , defined by the equalities $\varphi_{m, n}\left(\xi_{1}, \xi_{2}\right)=\xi_{1}^{m} \xi_{2}^{n}$ .

(a)

Compute the integrals

\int _{S^{3}}\left|\varphi _{m, n}\left(\xi _{1}, \xi _{2}\right)\right|^{2} d \mu _{3}\left(\xi _{1}, \xi _{2}\right) \quad (m, n=0,1,2, \ldots )

(b)

prove that the system $\left\{ \varphi_{m, n}\right\}_{m, n=0}^{\infty }$ is orthogonal;

(c)

prove that if the series $\sum_{m, n \geqslant 0} a_{m, n} \varphi_{m, n}$ converges uniformly on $\overline{B^{4}}$ and its sum equals $f$ , then the following generalization of the Cauchy integral formula holds:

f\left(\xi _{1}, \xi _{2}\right)=\frac{1}{2 \pi ^{2}} \int _{S^{3}} \frac{f\left(z_{1}, z_{2}\right)}{\left(1-\xi _{1} \bar{z}_{1}-\xi _{2} \bar{z}_{2}\right)^{2}} d \mu _{3}\left(z_{1}, z_{2}\right)

where $\left(\xi_{1}, \xi_{2}\right) \in B^{4}, B^{4}=\left\{ \left(\xi_{1}, \xi_{2}\right) \in \mathbb {C}^{2} \mid \left|\xi_{1}\right|^{2}+\left|\xi_{2}\right|^{2}<1\right\}$ .

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.