Math Problems Vault

Problem V.2.1

Evaluate the integral $\int_{0}^{+\infty } \int_{0}^{+\infty }|\ln x-\ln y| e^{-(x+y)} d x d y$ .

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Problem V.2.2

(a)(b)(c)(d)

Evaluate the integrals

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(a)

$\int_{(0,1)^{n}} \max \left(x_{1}, \ldots , x_{n}\right) d x_{1} \ldots d x_{n}$ ;

(b)

$\int_{(0,1)^{n}} \min \left(x_{1}, \ldots , x_{n}\right) d x_{1} \ldots d x_{n}$ ;

(c)

$\int_{(0,1)^{n}}\left(\frac{\min \left(x_{1}, \ldots , x_{n}\right)}{\max \left(x_{1}, \ldots , x_{n}\right)}\right)^{p} d x_{1} \ldots d x_{n} \quad (p>-1)$ ;

(d)

$\int_{[1,+\infty )^{n}} \frac{d x_{1} \ldots d x_{n}}{x_{1} \ldots x_{n}\left(\max \left(x_{1}, \ldots , x_{n}\right)\right)^{\varepsilon }} \quad (\varepsilon >0)$ .

Problem V.2.3

Evaluate the integral $\int_{\mathbb {R}^{n}} e^{-\| x\|^{2} / 2} d x$ .

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Problem V.2.4

(a)(b)

Evaluate the integrals

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(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 V.2.5

Evaluate the integral $\int_{\| y\| \leqslant 1} \frac{d y}{\| x-y\| } \quad \left(x, y \in \mathbb {R}^{3}\right)$ .

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Problem V.2.6

Evaluate the integral $\iiint \int_{x^{2}+y^{2}+u^{2}+v^{2} \leqslant 1} e^{x^{2}+y^{2}-u^{2}-v^{2}} d x d y d u d v$ .

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Problem V.2.7

Compute the integral $\int_{(A x, x) \leqslant 1} e^{(A x, x)} d x$ $\left(x \in \mathbb {R}^{4}, A\right.$ is a positive definite matrix).

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Problem V.2.8

Let $A=\left\{ \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb {R}^{4} \mid \sqrt{x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}<x_{1}\right\}$ . For which $t \in \mathbb {R}^{4}$ is the integral

K(t)=\int _{A} e^{-(x, t)} d x

finite, and what is it equal to?

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Problem V.2.9

On the segment $\left[a, b\right]$ two points are chosen at random. Find the mean value $M$ of the distance between them. What is the probability $P$ that the distance is greater than $M$ ? For which $m$ are the events "the distance is greater than $m$ " and "the distance is less than $m$ " equally probable?

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Remark.

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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.10

On the segment $\left[0, a\right]$ three numbers are chosen at random. What is the probability that they are the lengths of the sides of some triangle?

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Remark.

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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.11

(a)(b)

On the segment $\left[-a, a\right]$ two points $u, v$ are chosen at random.

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(a)

Which is more probable: that the roots of the equation $z^{2}+u z+v=0$ lie on the real axis (probability $P_{1}(a)$ ) or that the roots of this equation do not lie on the real axis (probability $P_{2}(a)$ )? What do the probabilities $P_{1}(a), P_{2}(a)$ tend to as $a \rightarrow +\infty$ ?

(b)

What is the probability that the biquadratic equation $z^{4}+u z^{2}+v=0$ has both real and complex roots?

Remark.

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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.12

In the unit disk two points are chosen at random. What is the mean value $S$ of the area of the circle constructed on the segment joining them as diameter?

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Remark.

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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.13

On a circle of radius $R$ two points are chosen at random. Find the mean value $L$ of the length of the chord joining them.

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Remark.

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When speaking of a random choice of points, we mean that the probability of choosing a point belonging to some set is proportional to its measure (length, area, volume).

Problem V.2.14

Let

A=\left\{ \left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n} \left\lvert \, \sum _{1 \leqslant k \leqslant n} \frac{x_{k}}{k} \leqslant 1\right., x_{1}, \ldots , x_{n} \geqslant 0\right\} .

Prove that for any $t \in \mathbb {R}$ the equality

\int \ldots \int e^{t\left(x_{1}+\ldots +x_{n}\right)} d x_{1} \ldots d x_{n}=\left(\frac{e^{t}-1}{t}\right)^{n}

holds.

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Problem V.2.15

Prove that for $m, n \in \mathbb {N}, m \leqslant n, a>0$ , the equality

\int _{\left[0,1\right]^{n}} \frac{d x}{\left(a+x_{1}+\ldots +x_{n}\right)^{m}}=(-1)^{m-1} \frac{\Delta _{n}\left(a^{n-m} \ln a\right)}{(m-1)!(n-m)!}

holds, where $\Delta_{1} f(a)=\Delta f(a)=f(a+1)-f(a), \quad \Delta_{n} f(a)=\Delta \left(\Delta_{n-1} f(a)\right)$ .

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Problem V.2.16

(a)(b)

Let $a_{0}, \ldots , a_{n}$ be positive numbers, $b_{0}=a_{1}$ , $b_{k}=a_{k+1}-a_{k}$ for $k=1, \ldots , n\left(a_{n+1}=a_{0}\right)$ ,

W_{n}=\left\{ \left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n} \mid \sum _{1 \leqslant k \leqslant n} x_{k}<1, x_{1}, \ldots , x_{n}>0\right\} .

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(a)

Prove that for any nonnegative and continuous function $f$ on $(0,+\infty )$ the equality

\begin{aligned} \int \ldots \int f\left(b_{0}+\sum _{k=1}^{n} b_{k} x_{1} \ldots x_{k}\right) x_{1}^{n-1} x_{2}^{n-2} \ldots x_{n-1} d x_{1} \ldots d x_{n} & = \\ & =\int \ldots \int f\left(a_{0}+\sum _{k=1}^{n}\left(a_{k}-a_{0}\right) x_{k}\right) d x_{1} \ldots d x_{n} \end{aligned}

holds.

(b)

Compute the integrals

\begin{gathered} \int \ldots \int _{(0,1)^{n}} \frac{x_{1}^{n-1} x_{2}^{n-2} \ldots x_{n-1} d x_{1} \ldots d x_{n}}{\left(b_{0}+b_{1} x_{1}+b_{2} x_{1} x_{2}+\ldots +b_{n} x_{1} \ldots x_{n}\right)^{n+1}} \\ \int \ldots \int \frac{d x_{1} \ldots d x_{n}}{\left(a_{0}+\left(a_{1}-a_{0}\right) x_{1}+\left(a_{2}-a_{0}\right) x_{2}+\ldots +\left(a_{n}-a_{0}\right) x_{n}\right)^{n+1}} . \end{gathered}

Problem V.2.17

(a)(b)

Prove that

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(a)

if the function $f$ is continuous and nonnegative on $\mathbb {R}$ , then for any real numbers $a$ and $b$ the equality

\frac{1}{\pi } \iint _{\mathbb {R}^{2}} \frac{f(a x+b y) d x d y}{\left(1+x^{2}\right)\left(1+y^{2}\right)}=\int _{-\infty }^{+\infty } f((\left|a\right|+\left|b\right|) u) \frac{d u}{1+u^{2}}

holds

(b)

for $0<p<1$ and $a=\left(a_{1}, \ldots , a_{n}\right) \in \mathbb {R}^{n}$ the equality $\left(x=\left(x_{1}, \ldots , x_{n}\right) \in \mathbb {R}^{n}\right)$

\int _{\mathbb {R}^{n}} \frac{\left|(a, x)\right|^{p} d x_{1} \ldots d x_{n}}{\left(1+x_{1}^{2}\right)\left(1+x_{2}^{2}\right) \ldots \left(1+x_{n}^{2}\right)}=\frac{\pi ^{n}}{\cos \frac{\pi }{2} p}\left(\sum _{1 \leqslant k \leqslant n}\left|a_{k}\right|\right)^{p} .

holds.

Problem V.2.18

Let $f \in C\left(\left[0,1\right]^{2}\right)$ . Find the limit

\lim _{n \rightarrow \infty } n^{2}\left(\iint _{\left[0,1\right]^{2}} f(x, y) d x d y-\frac{1}{n^{2}} \sum _{\substack {1 \leqslant k \leqslant n \\ 1 \leqslant j \leqslant n}} f\left(\frac{k-\frac{1}{2}}{n}, \frac{j-\frac{1}{2}}{n}\right)\right)

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Problem V.2.19

Let $p<1$ and

\begin{aligned} & \Phi (x, y)=\iint _{u^{2}+v^{2} \leqslant 1}\left((x-u)^{2}+(y-v)^{2}\right)^{-p / 2} d u d v \\ & \Psi (x, y)=\iint _{u^{2}+v^{2} \leqslant 1} \ln \left((x-u)^{2}+(y-v)^{2}\right) d u d v \end{aligned}

Prove that $\Phi , \Psi \in C^{1}\left(\mathbb {R}^{2}\right)$ , and find $\Psi$ .

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