Volume of the n-dimensional unit ball

For readers familiar with the gamma function $\Gamma$ : Prove that

for every positive integer $n$.

Find the volume of the $n$-dimensional ball

Find $\displaystyle \iint \cdots \int_{\Omega } x_{n}^{2}\, dx_{1}\, dx_{2}\cdots dx_{n}$, where the region $\Omega$ is determined by the inequalities:

Compute

Reduce the $n$-fold integral ($n\geqslant 2$) spread over the region $x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leqslant R^{2}$

to a single integral, where $f(u)$ is a continuous function.

Let

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

Find the volume of the $n$-dimensional ball $\sum_{i = 1}^{n} x_{i}^{2} \leqslant R^{2}$.

Find the volume of the $n$-dimensional cylinder $\sum_{i = 1}^{n-1} x_{i}^{2} \leqslant R^{2}, 0 \leqslant x_{n} \leqslant H$.

Find the volume of the $n$-dimensional cone $\sum_{i = 1}^{n-1} x_{i}^{2} \leqslant x^{2} x_{n}^{2}, \quad 0 \leqslant x_{n} \leqslant H$.

Compute the integral $\int_{G} x_{n}^{2} d x, x = \left(x_{1} ; \ldots ; x_{n}\right)$, where

Find the volume of the $n$-dimensional ellipsoid $\sum_{i = 1}^{n} \frac{x_{i}^{2}}{a_{i}^{2}} \leqslant 1$.

Compute the integral $\int_{\Omega }\left(R^{2}-x_{1}^{2}-\ldots -x_{n}^{2}\right)^{1 / 2} d x$, where $\Omega$ is the ball $\sum_{i = 1} x_{i}^{2} \leqslant R^{2}$.

Let the function $f(r)$ be continuous for $r \geqslant 0$. Reduce to a single integral $\int_{\Omega } f\left(\sqrt{x_{1}^{2}+\ldots +x_{n}^{2}}\right) d x$, where $\Omega$ is the ball $\sum_{i = 1}^{n} x_{i}^{2} \leqslant R^{2}$.

Let $V_{n}$ be the volume of the $n$-dimensional ball of radius $R$. Prove that:

Find the volume of a four-dimensional right circular:

Find the Lebesgue measure of the set

Prove that for $p>0$

where $C_{m}(p)=\lambda_{m}\left\{ \left(x_{1}, \ldots , x_{m}\right) \in \mathbb {R}^{m} \mid \left|x_{1}\right|^{p}+\ldots +\left|x_{m}\right|^{p} \leqslant 1\right\}$ (for the value of $C_{m}(p)$ see problem 3.36).

Let $E \subset \mathbb {R}^{n}$ be a bounded set, $0<\lambda_{n}(E)<\infty$, and let $M(E, \varepsilon )$ be the maximal number of $\varepsilon$-separated points lying in $E$. Prove that

where $E_{\varepsilon / 2}$ is the $(\varepsilon / 2)$-neighborhood of the set $E, \quad \alpha_{n}$ is the volume of the $n$-dimensional unit ball.

The thickness of a watermelon's rind equals $\frac{1}{20}$ of the radius of the watermelon (regarded as a ball). What percentage of the volume of the watermelon does the rind constitute

Let $A$ and $B$ be convex centrally symmetric (with center at zero) compacta in $\mathbb {R}^{n}$. Can one assert that $\lambda_{n}(A) \leqslant \lambda_{n}(B)$, if every central section of the set $A$ by a hyperplane has Lebesgue measure (of dimension $n-1$) smaller than the corresponding section of the set $B$? (Compare, for large $n$, the sections of the unit cube and of the ball having unit volume.)

Let $f$ be as in problem 10.3.7. Letting $\left\| \boldsymbol {x}\right\|$ denote the length of $\boldsymbol {x}$, show that

The $\bar\lambda_{k}$-measure of a $k$-dimensional ball with radius $r$ is $\frac{\pi^{k/2} r^{k}}{\Gamma \left(\left(k/2\right)+1\right)}$.

More generally than problem 10.3.10, the $k$-dimensional volume of the ellipsoid given by the equation $\sum_{i=1}^{k} \frac{x_{i}^{2}}{c_{i}^{2}}<1$ is $\frac{\pi^{k/2}}{\Gamma \left(\left(k/2\right)+1\right)} c_{1} \cdots c_{n}$. (Here $c_{1}, \ldots , c_{n}>0$.)