원기둥·구면좌표 삼중적분

In the following exercises, evaluate the triple integrals $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)=z,$ $B=\{(x,y,z)|{x}^{2}+{y}^{2}\le 9,x\ge 0,y\ge 0,0\le z\le 1\}$

In the following exercises, evaluate the triple integrals $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)=x{z}^{2},$ $B=\{(x,y,z)|{x}^{2}+{y}^{2}\le 16,x\ge 0,y\le 0,-1\le z\le 1\}$

In the following exercises, evaluate the triple integrals $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)=xy,$ $B=\{(x,y,z)|{x}^{2}+{y}^{2}\le 1,x\ge 0,y\ge 0,x\le y,-1\le z\le 1\}$

In the following exercises, evaluate the triple integrals $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)={x}^{2}+{y}^{2},$ $B=\{(x,y,z)|{x}^{2}+{y}^{2}\le 4,x\ge 0,x\le y,0\le z\le 3\}$

In the following exercises, evaluate the triple integrals $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)={e}^{\sqrt{{x}^{2}+{y}^{2}}},$ $B=\{(x,y,z)|1\le {x}^{2}+{y}^{2}\le 4,y\le 0,x\le y\sqrt{3},2\le z\le 3\}$

In the following exercises, evaluate the triple integrals $\iiint_{B}f(x,y,z)dV$ over the solid $B.$

$f(x,y,z)=\sqrt{{x}^{2}+{y}^{2}},$ $B=\{(x,y,z)|1\le {x}^{2}+{y}^{2}\le 9,y\le 0,0\le z\le 1\}$

a. Let $B$ be a cylindrical shell with inner radius $a,$ outer radius $b,$ and height $c,$ where $0<a<b$ and $c>0.$ Assume that a function $F$ defined on $B$ can be expressed in cylindrical coordinates as $F(x,y,z)=f(r)+h(z),$ where $f$ and $h$ are differentiable functions. If $\int_{a}^{b}\tilde{f}(r)dr=0$ and $\tilde{h}(0)=0,$ where $\tilde{f}$ and $\tilde{h}$ are antiderivatives of $f$ and $h,$ respectively, show that $$\iiint_{B}F(x,y,z)dV=2\pi c\left(b\tilde{f}(b)-a\tilde{f}(a)\right)+\pi ({b}^{2}-{a}^{2})\tilde{h}(c).$$ b. Use the previous result to show that $\iiint_{B}\left(z+\sin \sqrt{{x}^{2}+{y}^{2}}\right)dx\,dy\,dz=6{\pi }^{2}(\pi -2),$ where $B$ is a cylindrical shell with inner radius $\pi ,$ outer radius $2\pi ,$ and height $2.$

a. Let $B$ be a cylindrical shell with inner radius $a,$ outer radius $b,$ and height $c,$ where $0<a<b$ and $c>0.$ Assume that a function $F$ defined on $B$ can be expressed in cylindrical coordinates as $F(x,y,z)=f(r)g(\theta )h(z),$ where $f,g,\,\text{and}\,h$ are differentiable functions. If $\int_{a}^{b}\tilde{f}(r)dr=0,$ where $\tilde{f}$ is an antiderivative of $f,$ show that $$\iiint_{B}F(x,y,z)dV=\left[b\tilde{f}(b)-a\tilde{f}(a)\right]\,\left[\tilde{g}(2\pi )-\tilde{g}(0)\right]\,\left[\tilde{h}(c)-\tilde{h}(0)\right],$$

where $\tilde{g}$ and $\tilde{h}$ are antiderivatives of $g$ and $h,$ respectively. b. Use the previous result to show that $\iiint_{B}z\,\sin \sqrt{{x}^{2}+{y}^{2}}dx\,dy\,dz=-12{\pi }^{2},$ where $B$ is a cylindrical shell with inner radius $\pi ,$ outer radius $2\pi ,$ and height $2.$

$E$ is inside the right circular cylinder $r=4\,\sin \,\theta ,$ above the $r\theta$-plane, and inside the sphere ${r}^{2}+{z}^{2}=16.$

$E$ is inside the right circular cylinder $r=\cos \,\theta ,$ above the $r\theta$-plane, and inside the sphere ${r}^{2}+{z}^{2}=9.$

$E$ is located in the first octant and is bounded by the circular paraboloid $z=9-3{r}^{2},$ the cylinder $r=\sqrt{3},$ and the plane $r(\cos \,\theta +\sin \,\theta )=20-z.$

$E$ is located in the first octant outside the circular paraboloid $z=10-2{r}^{2}$ and inside the cylinder $r=\sqrt{5}$ and is bounded also by the planes $z=20$ and $\theta =\frac{\pi }{4}.$

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in cylindrical coordinates. Let $f(r,\theta ,z)$ be the corresponding function in cylindrical coordinates. b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to cylindrical coordinates and evaluate it.

$g(x,y,z)=\frac{1}{x+3},$ $E=\{(x,y,z)|0\le {x}^{2}+{y}^{2}\le 9,x\ge 0,y\ge 0,0\le z\le x+3\}$

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in cylindrical coordinates. Let $f(r,\theta ,z)$ be the corresponding function in cylindrical coordinates. b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to cylindrical coordinates and evaluate it.

$g(x,y,z)={x}^{2}+{y}^{2},$ $E=\{(x,y,z)|0\le {x}^{2}+{y}^{2}\le 4,y\ge 0,0\le z\le 3-x\}$

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in cylindrical coordinates. Let $f(r,\theta ,z)$ be the corresponding function in cylindrical coordinates. b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to cylindrical coordinates and evaluate it.

$g(x,y,z)=x,$ $E=\{(x,y,z)|1\le {y}^{2}+{z}^{2}\le 9,0\le x\le 9-{y}^{2}-{z}^{2}\}$

In the following exercises, the function $g$ and region $E$ are given in rectangular coordinates.

a. Express the region $E$ and the function $g$ in cylindrical coordinates. Let $f(r,\theta ,z)$ be the corresponding function in cylindrical coordinates. b. Convert the integral $\iiint_{E}g(x,y,z)dV$ to cylindrical coordinates and evaluate it.

$g(x,y,z)=y,$ $E=\{(x,y,z)|1\le {x}^{2}+{z}^{2}\le 9,0\le y\le 9-{x}^{2}-{z}^{2}\}$

$E$ is above the $xy$-plane, inside the cylinder ${x}^{2}+{y}^{2}=1,$ and below the plane $z=1.$

$E$ is below the plane $z=1$ and inside the paraboloid $z={x}^{2}+{y}^{2}.$

In the following exercises, find the volume of the solid $E$ whose boundaries are given in rectangular coordinates.

$E$ is bounded by the circular cone $z=\sqrt{{x}^{2}+{y}^{2}}$ and $z=1.$

$E$ is located above the $xy$-plane, below $z=1,$ outside the one-sheeted hyperboloid ${x}^{2}+{y}^{2}-{z}^{2}=1,$ and inside the cylinder ${x}^{2}+{y}^{2}=2.$