Math Problems Vault

Problem 4161

Investigate the convergence of the following improper integral having an unbounded domain of integration ( $0<m\leqslant \left|\varphi (x,y)\right|\leqslant M$ ):

\iint \limits _{x^{2}+y^{2}\geqslant 1} \dfrac {\varphi (x,y)}{(x^{2}+y^{2})^{p}}\, dx\, dy .

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Problem 4162

Investigate the convergence of the following improper integral having an unbounded domain of integration ( $0<m\leqslant \left|\varphi (x,y)\right|\leqslant M$ ):

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } \dfrac {dx\, dy}{(1+\left|x\right|^{p})(1+\left|y\right|^{q})} .

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Problem 4163

Investigate the convergence of the following improper integral having an unbounded domain of integration ( $0<m\leqslant \left|\varphi (x,y)\right|\leqslant M$ ):

\iint \limits _{0\leqslant y\leqslant 1} \dfrac {\varphi (x,y)}{(1+x^{2}+y^{2})^{p}}\, dx\, dy .

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Problem 4164

Investigate the convergence of the following improper integral having an unbounded domain of integration:

\iint \limits _{\left|x\right|+\left|y\right|\geqslant 1} \dfrac {dx\, dy}{\left|x\right|^{p}+\left|y\right|^{q}} \quad (p>0,\ q>0) .

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Problem 4165

Investigate the convergence of the following improper integral having an unbounded domain of integration:

\iint \limits _{x+y\geqslant 1} \dfrac {\sin x\sin y}{(x+y)^{p}}\, dx\, dy .

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Problem 4166

Prove: if the continuous function $f(x,y)$ is non-negative and $S_{n}$ $(n=1,2,\ldots )$ is any sequence of bounded closed regions that exhausts the region $S$ , then

\iint \limits _{S} f(x,y)\, dx\, dy = \lim _{n\to \infty } \iint \limits _{S_{n}} f(x,y)\, dx\, dy ,

where the left-hand side and the right-hand side are simultaneously meaningful or simultaneously meaningless.

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Problem 4167

Prove that:

\lim _{n\to \infty } \iint \limits _{\substack {\left|x\right|\leqslant n \\ \left|y\right|\leqslant n}} \sin (x^{2}+y^{2})\, dx\, dy = \pi ,

but

\lim _{n\to \infty } \iint \limits _{x^{2}+y^{2}\leqslant 2n\pi } \sin (x^{2}+y^{2})\, dx\, dy = 0

( $n$ is a natural number).

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Problem 4168

Prove that although the iterated integrals

\int _{1}^{+\infty } dx \int _{1}^{+\infty } \dfrac {x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}\, dy

and

\int _{1}^{+\infty } dy \int _{1}^{+\infty } \dfrac {x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}\, dx

converge, the integral

\iint \limits _{x>1,\ y>1} \dfrac {x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}\, dx\, dy

diverges.

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Problem 4169

Compute the following integral:

\iint \limits _{\substack {xy\geqslant 1 \\ x\geqslant 1}} \dfrac {dx\, dy}{x^{p}y^{q}} .

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Problem 4170

Compute the following integral:

\iint \limits _{\substack {x+y\geqslant 1 \\ 0\leqslant x\leqslant 1}} \dfrac {dx\, dy}{(x+y)^{p}} .

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Problem 4171

Compute the following integral:

\iint \limits _{x^{2}+y^{2}\leqslant 1} \dfrac {dx\, dy}{\sqrt{1-x^{2}-y^{2}}} .

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Problem 4172

Compute the following integral:

\iint \limits _{x^{2}+y^{2}\geqslant 1} \dfrac {dx\, dy}{(x^{2}+y^{2})^{p}} .

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Problem 4173

Compute the following integral:

\iint \limits _{y\geqslant x^{2}+1} \dfrac {dx\, dy}{x^{4}+y^{2}} .

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Problem 4174

Compute the following integral:

\iint \limits _{0\leqslant x\leqslant y} e^{-(x+y)}\, dx\, dy .

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Problem 4175

Transform to polar coordinates to compute the integral:

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{-(x^{2}+y^{2})}\, dx\, dy .

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Problem 4176

Transform to polar coordinates to compute the integral:

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{-(x^{2}+y^{2})}\cos (x^{2}+y^{2})\, dx\, dy .

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Problem 4177

Transform to polar coordinates to compute the integral:

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{-(x^{2}+y^{2})}\sin (x^{2}+y^{2})\, dx\, dy .

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Problem 4178

Compute the integral:

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{ax^{2}+2bxy+cy^{2}+2dx+2ey+f}\, dx\, dy ,

where $a<0$ , $ac-b^{2}>0$ .

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Problem 4179

Compute the following integral:

\iint \limits _{\substack {x^{2}/a^{2}+y^{2}/b^{2}\geqslant 1}} e^{-\left(x^{2}/a^{2}+y^{2}/b^{2}\right)}\, dx\, dy .

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Problem 4180

Compute the following integral ( $0<\left|\varepsilon \right|<1$ ):

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } xy\, e^{-\left(x^{2}/a^{2}+2\varepsilon \frac{x}{a}\cdot \frac{y}{b}+y^{2}/b^{2}\right)}\, dx\, dy \quad (0<\left|\varepsilon \right|<1) .

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Problem 4181

Investigate the convergence of the improper double integral of a discontinuous function ( $0<m\leqslant \left|\varphi (x,y)\right|\leqslant M$ ):

\iint \limits _{\Omega } \dfrac {dx\, dy}{x^{2}+y^{2}} ,

where the domain $\Omega$ is determined by the conditions $\left|y\right|\leqslant x^{2}$ , $x^{2}+y^{2}\leqslant 1$ .

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Problem 4182

Investigate the convergence of the improper double integral of a discontinuous function ( $0<m\leqslant \left|\varphi (x,y)\right|\leqslant M$ ):

\iint \limits _{x^{2}+y^{2}\leqslant 1} \dfrac {\varphi (x,y)}{(x^{2}+xy+y^{2})^{p}}\, dx\, dy .

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Problem 4183

Investigate the convergence of the following integral ( $p>0$ , $q>0$ ):

\iint \limits _{\left|x\right|+\left|y\right|\leqslant 1} \dfrac {dx\, dy}{\left|x\right|^{p}+\left|y\right|^{q}} .

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Problem 4184

Investigate the convergence of the improper double integral of a discontinuous function ( $0<m\leqslant \left|\varphi (x,y)\right|\leqslant M$ ):

\int _{0}^{a}\int _{0}^{a} \dfrac {\varphi (x,y)}{\left|x-y\right|^{p}}\, dx\, dy .

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Problem 4185

Investigate the convergence of the improper double integral of a discontinuous function ( $0<m\leqslant \left|\varphi (x,y)\right|\leqslant M$ ):

\iint \limits _{x^{2}+y^{2}\leqslant 1} \dfrac {\varphi (x,y)}{(1-x^{2}-y^{2})^{p}}\, dx\, dy .

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Problem 4186

Prove that if: 1) the function $\varphi (x,y)$ is continuous on the bounded domain $a\leqslant x\leqslant A$ , $b\leqslant y\leqslant B$ ; 2) the function $f(x)$ is continuous on the closed interval $a\leqslant x\leqslant A$ ; 3) $p<1$ , then the integral

\int _{a}^{A} dx \int _{b}^{B} \dfrac {\varphi (x,y)}{\left|f(x)-y\right|^{p}}\, dy

converges.

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Problem 4187

Compute the following integral:

\iint \limits _{x^{2}+y^{2}\leqslant 1} \ln \dfrac {1}{\sqrt{x^{2}+y^{2}}}\, dx\, dy .

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Problem 4188

Compute the following integral ( $a>0$ ):

\int _{0}^{a} dx \int _{0}^{x} \dfrac {dy}{\sqrt{(a-x)(x-y)}} .

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Problem 4189

Compute the integral $\displaystyle \iint \limits_{\Omega } \ln \sin (x-y)\, dx\, dy$ , where the domain $\Omega$ is bounded by the lines $y=0$ , $y=x$ , $x=\pi$ .

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Problem 4190

Compute the integral:

\iint \limits _{x^{2}+y^{2}\leqslant x} \dfrac {dx\, dy}{\sqrt{x^{2}+y^{2}}} .

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Problem 4191

Investigate the convergence of the following triple integral ( $0<m\leqslant \left|\varphi (x,y,z)\right|\leqslant M$ ):

\iiint \limits _{x^{2}+y^{2}+z^{2}\geqslant 1} \dfrac {\varphi (x,y,z)}{(x^{2}+y^{2}+z^{2})^{p}}\, dx\, dy\, dz .

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Problem 4192

Study the convergence of the following triple integral ( $0<m\leqslant \left|\varphi (x,y,z)\right|\leqslant M$ ):

\iiint \limits _{x^{2}+y^{2}+z^{2}\leqslant 1} \dfrac {\varphi (x,y,z)}{(x^{2}+y^{2}+z^{2})^{p}}\, dx\, dy\, dz .

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Problem 4193

Study the convergence of the following triple integral ( $p>0$ , $q>0$ , $r>0$ ):

\iiint \limits _{\left|x\right|+\left|y\right|+\left|z\right|\geqslant 1} \dfrac {dx\, dy\, dz}{\left|x\right|^{p}+\left|y\right|^{q}+\left|z\right|^{r}} .

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Problem 4194

Study the convergence of the triple improper integral of a discontinuous function:

\int _{0}^{a}\int _{0}^{a}\int _{0}^{a} \dfrac {f(x,y,z)\, dx\, dy\, dz}{\left\{ [y-\varphi (x)]^{2}+[z-\psi (x)]^{2}\right\} ^{p}} ,

where $0<m\leqslant \left|f(x,y,z)\right|\leqslant M$ , and $\varphi (x)$ and $\psi (x)$ are continuous functions on the closed interval $[0,a]$ .

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Problem 4195

Study the convergence of the following triple integral:

\iiint \limits _{\substack {\left|x\right|\leqslant 1 \\ \left|y\right|\leqslant 1 \\ \left|z\right|\leqslant 1}} \dfrac {dx\, dy\, dz}{\left|x+y-z\right|^{p}} .

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Problem 4196

Compute the integral:

\int _{0}^{1}\int _{0}^{1}\int _{0}^{1} \dfrac {dx\, dy\, dz}{x^{p}y^{q}z^{r}} .

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Problem 4197

Compute the integral:

\iiint \limits _{x^{2}+y^{2}+z^{2}\geqslant 1} \dfrac {dx\, dy\, dz}{(x^{2}+y^{2}+z^{2})^{3}} .

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Problem 4198

Compute the integral:

\iiint \limits _{x^{2}+y^{2}+z^{2}\leqslant 1} \dfrac {dx\, dy\, dz}{(1-x^{2}-y^{2}-z^{2})^{p}} .

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Problem 4199

Compute the integral:

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{-(x^{2}+y^{2}+z^{2})}\, dx\, dy\, dz .

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Problem 4200

Compute the integral

\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty } e^{-P(x_{1},x_{2},x_{3})}\, dx_{1}\, dx_{2}\, dx_{3} ,

where $P(x_{1},x_{2},x_{3}) = \displaystyle \sum_{i=1}^{3}\sum_{j=1}^{3} a_{ij}x_{i}x_{j}$ ( $a_{ij}=a_{ji}$ ) is a positive definite form.

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