### Triple Integrals In Cylindrical And Spherical Coordinates

(requires JavaScript)

1. Sketch the solid whose volume is given by the integral ${\int }_{0}^{4}{\int }_{0}^{2\pi }{\int }_{r}^{4}r\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}d\theta \phantom{\rule{0.2em}{0ex}}dr$ and evaluate the integral.
$\frac{64\pi }{3}$
2. Set up the triple integral of an arbitrary continuous function $f\left(x,y,z\right)$ in cylindrical or spherical coordinates over the solid shown in the figure.

${\int }_{0}^{\pi /2}{\int }_{0}^{3}{\int }_{0}^{2}rf\left(r\mathrm{cos}\left(\theta \right),r\mathrm{sin}\left(\theta \right),z\right)\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\theta$
3. Evaluate $\underset{E}{\iiint }{x}^{2}\phantom{\rule{0.2em}{0ex}}dV$, where $E$ is the solid that lies within the cylinder ${x}^{2}+{y}^{2}=1$, above the plane $z=0$, and below the cone ${z}^{2}=4{x}^{2}+4{y}^{2}$.
$2\pi /5$
4. Evaluate the integral ${\int }_{-3}^{3}{\int }_{0}^{\sqrt{9-{x}^{2}}}{\int }_{0}^{9-{x}^{2}-{y}^{2}}\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx$ by changing to cylindrical coordinates.
5. Sketch the solid whose volume is given by the integral ${\int }_{0}^{\pi /6}{\int }_{0}^{\pi /2}{\int }_{0}^{3}{\rho }^{2}\mathrm{sin}\left(\phi \right)\phantom{\rule{0.2em}{0ex}}d\rho \phantom{\rule{0.2em}{0ex}}d\theta \phantom{\rule{0.2em}{0ex}}d\phi$ and evaluate the integral.
$\frac{9\pi }{4}\left(2-\sqrt{3}\right)$
6. Set up the triple integral of an arbitrary continuous function $f\left(x,y,z\right)$ in cylindrical or spherical coordinates over the solid between the spheres ${x}^{2}+{y}^{2}+{z}^{2}=1$ and ${x}^{2}+{y}^{2}+{z}^{2}=4$ in all octants above the $xy\text{-plane}$ except for the first octant. Sketch the solid.

7. Use spherical coordinates to evaluate $\underset{H}{\iiint }\left({x}^{2}+{y}^{2}\right)\phantom{\rule{0.2em}{0ex}}dV$, where $H$ is the region that lies above the $xy\text{-plane}$ and below the sphere ${x}^{2}+{y}^{2}+{z}^{2}=1$.
8. Use spherical coordinates to evaluate $\underset{E}{\iiint }z\phantom{\rule{0.2em}{0ex}}dV$, where $E$ lies between the spheres ${x}^{2}+{y}^{2}+{z}^{2}=1$ and ${x}^{2}+{y}^{2}+{z}^{2}=4$ in the first octant.
$\frac{15\pi }{16}$
9. Use spherical coordinates to evaluate $\underset{E}{\iiint }{e}^{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}\phantom{\rule{0.2em}{0ex}}dV$, where $E$ is enclosed by the sphere ${x}^{2}+{y}^{2}+{z}^{2}=9$ in the first octant.
10. Find the volume and centroid of the solid $E$ that lies above the cone $z=\sqrt{{x}^{2}+{y}^{2}}$ and below the sphere ${x}^{2}+{y}^{2}+{z}^{2}=1$.
$\frac{2\pi }{3}\left(1-\frac{1}{\sqrt{2}}\right)$ and $\left(0,0,\frac{3}{8\left(2-\sqrt{2}\right)}\right)$
11. Evaluate the integral ${\int }_{0}^{1}{\int }_{0}^{\sqrt{1-{x}^{2}}}{\int }_{\sqrt{{x}^{2}+{y}^{2}}}^{\sqrt{2-{x}^{2}-{y}^{2}}}xy\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx$ by changing to spherical coordinates.
$\frac{4\sqrt{2}-5}{15}$