원기둥·구면좌표 삼중적분

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

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

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

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

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 $B.$

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

$E$ 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 $z={x}^{2}+{y}^{2}.$

$E$ 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 ${x}^{2}+{y}^{2}=2.$

$E$ is located inside the cylinder ${x}^{2}+{y}^{2}=1$ and between the circular paraboloids $z=1-{x}^{2}-{y}^{2}$ and $z={x}^{2}+{y}^{2}.$

$E$ is located inside the sphere ${x}^{2}+{y}^{2}+{z}^{2}=1,$ above the $xy$-plane, and inside the circular cone $z=\sqrt{{x}^{2}+{y}^{2}}.$

$E$ is located inside the circular cone ${x}^{2}+{y}^{2}={(z-1)}^{2}$ and between the planes $z=0$ and $z=2.$

$E$ is located inside the cylinder ${x}^{2}+{y}^{2}=1$, above the circular cone $z=1-\sqrt{{x}^{2}+{y}^{2}}$, and below the circular paraboloid $z=1+{x}^{2}+{y}^{2}$, and between the planes $z=0$ and $z=2$.

Convert the integral $\int_{0}^{1}\,\int_{-\sqrt{1-{y}^{2}}}^{\sqrt{1-{y}^{2}}}\,\int_{{x}^{2}+{y}^{2}}^{\sqrt{{x}^{2}+{y}^{2}}}xz\,dz\,dx\,dy$ into an integral in cylindrical coordinates.