Triple Integrals In Cylindrical And Spherical Coordinates

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1. Sketch the solid whose volume is given by the integral ${\displaystyle {\int }_0^4{\int }_0^{2\pi }{\int }_r^{4}rdzd\theta dr}$ and evaluate the integral.
   ${\displaystyle \frac{64\pi }{3}}$

2. Set up the triple integral of an arbitrary continuous function $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)$ in cylindrical or spherical coordinates over the solid shown in the figure.

   

   ${\displaystyle {\int }_0^{\pi /2}{\int }_0^3{\int }_0^{2}rf<!--FUNCTION\; APPLICATION-->\left(r\cos <!--FUNCTION\; APPLICATION-->\left(\theta \right),r\sin <!--FUNCTION\; APPLICATION-->\left(\theta \right),z\right)dzdrd\theta }$
3. Evaluate ${\displaystyle \underset{E}{\iiint }{x}^{2}dV}$, where $E$ is the solid that lies within the cylinder $x^2+y^2=1$, above the plane $z=0$, and below the cone $z^2=4x^2+4y^2$.
   $2\pi /5$
4. Evaluate the integral ${\displaystyle {\int }_{-3}^3{\int }_0^{\sqrt{9-x^2}}{\int }_0^{9-x^2-y^2}\sqrt{x^2+y^2}dzdydx}$ by changing to cylindrical coordinates.
5. Sketch the solid whose volume is given by the integral ${\displaystyle {\int }_0^{\pi /6}{\int }_0^{\pi /2}{\int }_0^3{\rho }^2\sin <!--FUNCTION\; APPLICATION-->\left(\phi \right)d\rho d\theta d\phi }$ and evaluate the integral.
   ${\displaystyle \frac{9\pi }{4}\left(2-\sqrt{3}\right)}$

6. Set up the triple integral of an arbitrary continuous function $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)$ 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 ${\displaystyle \underset{H}{\iiint }\left(x^2+y^2\right)dV}$, 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 ${\displaystyle \underset{E}{\iiint }zdV}$, 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.
   ${\displaystyle \frac{15\pi }{16}}$
9. Use spherical coordinates to evaluate ${\displaystyle \underset{E}{\iiint }{e}^{\sqrt{x^2+y^2+z^2}}dV}$, 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 $x^2+y^2+z^2=1$.
    ${\displaystyle \frac{2\pi }{3}\left(1-\frac{1}{\sqrt{2}}\right)}$ and ${\displaystyle \left(0,0,\frac{3}{8\left(2-\sqrt{2}\right)}\right)}$
11. Evaluate the integral ${\displaystyle {\int }_0^1{\int }_0^{\sqrt{1-x^2}}{\int }_{\sqrt{x^2+y^2}}^{\sqrt{2-x^2-y^2}}xydzdydx}$ by changing to spherical coordinates.
    ${\displaystyle \frac{4\sqrt{2}-5}{15}}$