Math Problems Vault

Problem 4005

Sketch a solid whose volume equals the integral

V = \int _{0}^{1} dx \int _{0}^{1-x} \left(x^{2}+y^{2}\right)\, dy .

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Problem 4006

Sketch the volumes represented by the following double integrals:

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(a)

$\displaystyle \iint \limits_{\substack {0<x+y<1 \\ x\geqslant 0,y\geqslant 0}} (x+y)\, dx\, dy$ ;

(b)

$\displaystyle \iint \limits_{\frac{x^{2}}{4}+\frac{y^{2}}{9}\leqslant 1} \sqrt{1-\dfrac {x^{2}}{4}-\dfrac {y^{2}}{9}}\, dx\, dy$ ;

(c)

$\displaystyle \iint \limits_{\left|x\right|+\left|y\right|\leqslant 1} \left(x^{2}+y^{2}\right)\, dx\, dy$ ;

(d)

$\displaystyle \iint \limits_{x^{2}+y^{2}\leqslant x} \sqrt{x^{2}+y^{2}}\, dx\, dy$ ;

(e)

$\displaystyle \iint \limits_{\substack {1\leqslant x\leqslant 2 \\ x\leqslant y\leqslant 2x}} \sqrt{xy}\, dx\, dy$ ;

(f)

$\displaystyle \iint \limits_{x^{2}+y^{2}\leqslant 1} \sin \pi \sqrt{x^{2}+y^{2}}\, dx\, dy$ .

Problem 4007

Find the volume bounded by the following surfaces: $z=1+x+y$ , $z=0$ , $x+y=1$ , $x=0$ , $y=0$ .

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Problem 4008

Find the volume bounded by the following surfaces: $x+y+z=a$ , $x^{2}+y^{2}=R^{2}$ , $x=0$ , $y=0$ , $z=0$ $(a\geqslant R\sqrt{2})$ .

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Problem 4009

Find the volume bounded by the following surfaces: $z=x^{2}+y^{2}$ , $y=x^{2}$ , $y=1$ , $z=0$ .

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Problem 4010

Find the volume bounded by the following surfaces: $z=\cos x\cos y$ , $z=0$ , $\left|x+y\right|\leqslant \dfrac {\pi }{2}$ , $\left|x-y\right|\leqslant \dfrac {\pi }{2}$ .

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Problem 4011

Find the volume bounded by the following surfaces: $z=\sin \dfrac {\pi y}{2x}$ , $z=0$ , $y=x$ , $y=0$ , $x=\pi$ .

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Problem 4012

Find the volume bounded by the following surfaces: $z=xy$ , $x+y+z=1$ , $z=0$ .

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Problem 4013

Transform to polar coordinates to find the volume bounded by the following surfaces: $z^{2}=xy$ , $x^{2}+y^{2}=a^{2}$ .

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Problem 4014

Find the volume bounded by the following surfaces: $z=x+y$ , $\left(x^{2}+y^{2}\right)^{2}=2xy$ , $z=0$ $(x\geqslant 0,y\geqslant 0)$ .

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Problem 4015

Find the volume bounded by the following surfaces: $z=x^{2}+y^{2}$ , $x^{2}+y^{2}=x$ , $x^{2}+y^{2}=2x$ , $z=0$ .

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Problem 4016

Find the volume bounded by the following surfaces: $x^{2}+y^{2}+z^{2}=a^{2}$ , $x^{2}+y^{2}\geqslant a\left|x\right|$ $(a>0)$ .

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Problem 4017

Find the volume bounded by the following surfaces: $x^{2}+y^{2}-az=0$ , $\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right)$ , $z=0$ $(a>0)$ .

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Problem 4018

Find the volume bounded by the following surfaces: $z=e^{-\left(x^{2}+y^{2}\right)}$ , $z=0$ , $x^{2}+y^{2}=R^{2}$ .

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Problem 4019

Find the volume bounded by the following surfaces: $z=c\cos \dfrac {\pi \sqrt{x^{2}+y^{2}}}{2a}$ , $x^{2}+y^{2}=a^{2}$ , $y=x\operatorname {tg}\alpha$ , $y=x\operatorname {tg}\beta$ $(a>0,c>0,0\leqslant \alpha <\beta \leqslant 2\pi )$ .

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Problem 4020

Find the volume bounded by the following surfaces: $z=x^{2}+y^{2}$ , $z=x+y$ .

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Problem 4021

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

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Problem 4022

Find the volume bounded by the following surfaces (parameters assumed positive): $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}-\dfrac {z^{2}}{c^{2}}=-1$ , $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ .

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Problem 4023

Find the volume bounded by the following surfaces (parameters assumed positive): $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {z}{c}$ , $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=\dfrac {x}{a}+\dfrac {y}{b}$ , $z=0$ .

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Problem 4024

Find the volume bounded by the following surfaces (parameters assumed positive): $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{2}+\dfrac {z}{c}=1$ , $z=0$ .

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Problem 4025

Find the volume bounded by the following surfaces (parameters assumed positive): $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{2}+\dfrac {z^{2}}{c^{2}}=1$ , $x=0$ , $y=0$ , $z=0$ .

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Problem 4026

Find the volume bounded by the following surfaces (parameters assumed positive): $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , $\left(\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}\right)^{2}=\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}$ .

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Problem 4027

Find the volume bounded by the following surfaces (parameters assumed positive): $z^{2}=xy$ , $x+y=a$ , $x+y=b$ $(0<a<b)$ .

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Problem 4028

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

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Problem 4029

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

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Problem 4030

Find the volume bounded by the following surfaces (parameters assumed positive): $z=c\sin \dfrac {\pi xy}{a^{2}}$ , $z=0$ , $xy=a^{2}$ , $y=\alpha x$ , $y=\beta x$ $(0<\alpha <\beta ;x\geqslant 0)$ .

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Problem 4031

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

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Problem 4032

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

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Problem 4033

Find the volume bounded by the following surfaces (parameters assumed positive): $z=c\operatorname {arctg}\dfrac {y}{x}$ , $z=0$ , $\sqrt{x^{2}+y^{2}}=a\operatorname {arctg}\dfrac {y}{x}$ $(y\geqslant 0)$ .

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Problem 4033.1

[NO TASK]

Problem 4034

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

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Problem 4035

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

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