5.5 Triple integrals in cylindrical and spherical coordinates  (Page 10/12)

$f\left(x,y,z\right)=z;$ $E=\left\{\left(x,y,z\right)|0\le x^2+y^2+z^2\le 1,z\ge 0\right\}$

$f\left(x,y,z\right)=x+y;$ $E=\left\{\left(x,y,z\right)|1\le x^2+y^2+z^2\le 2,z\ge 0,y\ge 0\right\}$

$f\left(x,y,z\right)=2xy;$ $E=\left\{\left(x,y,z\right)|\sqrt{x^2+y^2}\le z\le \sqrt{1-x^2-y^2},x\ge 0,y\ge 0\right\}$

$f\left(x,y,z\right)=z;$ $E=\left\{\left(x,y,z\right)|x^2+y^2+z^2-2z\le 0,\sqrt{x^2+y^2}\le z\right\}$

$E=\left\{\left(x,y,z\right)|\sqrt{x^2+y^2}\le z\le \sqrt{16-x^2-y^2},x\ge 0,y\ge 0\right\}$

$E=\left\{\left(x,y,z\right)|x^2+y^2+z^2-2z\le 0,\sqrt{x^2+y^2}\le z\right\}$

Use spherical coordinates to find the volume of the solid situated outside the sphere $\rho =1$ and inside the sphere $\rho =\text{cos}\phi ,$ with $\phi \in \left[0,\frac{\pi }{2}\right].$

Use spherical coordinates to find the volume of the ball $\rho \le 3$ that is situated between the cones $\phi =\frac{\pi }{4}\text{and}\phi =\frac{\pi }{3}.$

Convert the integral ${\displaystyle \underset{\mathrm{-4}}{\overset{4}{\int }}{\displaystyle \underset{\text{−}\sqrt{16-y^2}}{\overset{\sqrt{16-y^2}}{\int }}{\displaystyle \underset{\text{−}\sqrt{16-x^2-y^2}}{\overset{\sqrt{16-x^2-y^2}}{\int }}\left(x^2+y^2+z^2\right)}}}dzdxdy$ into an integral in spherical coordinates.

Convert the integral ${\displaystyle \underset{0}{\overset{4}{\int }}{\displaystyle \underset{0}{\overset{\sqrt{16-x^2}}{\int }}{\displaystyle \underset{\text{−}\sqrt{16-x^2-y^2}}{\overset{\sqrt{16-x^2-y^2}}{\int }}{\left(x^2+y^2+z^2\right)}^2}}}dzdydx$ into an integral in spherical coordinates.

Convert the integral ${\displaystyle \underset{\mathrm{-2}}{\overset{2}{\int }}{\displaystyle \underset{\text{−}\sqrt{4-x^2}}{\overset{\sqrt{4-x^2}}{\int }}{\displaystyle \underset{\sqrt{x^2+y^2}}{\overset{\sqrt{16-x^2-y^2}}{\int }}dzdydx}}}$ into an integral in spherical coordinates and evaluate it.

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates ${\displaystyle \underset{\pi \text{/}2}{\overset{\pi }{\int }}{\displaystyle \underset{5\pi \text{/}6}{\overset{\pi \text{/}6}{\int }}{\displaystyle \underset{0}{\overset{2}{\int }}{\rho }^2\text{sin}\phi }}}d\rho d\phi d\theta .$ Find the volume $V$ of the solid. Round your answer to three decimal places.

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as ${\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{3\pi \text{/}4}{\overset{\pi \text{/}4}{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}{\rho }^2\text{sin}\phi }}}d\rho d\phi d\theta .$ Find the volume $V$ of the solid. Round your answer to three decimal places.

[T] Use a CAS to evaluate the integral ${\displaystyle \underset{E}{\iiint }\left(x^2+y^2\right)}dV$ where $E$ lies above the paraboloid $z=x^2+y^2$ and below the plane $z=3y.$

[T]

1. Evaluate the integral ${\displaystyle \underset{E}{\iiint }{e}^{\sqrt{x^2+y^2+z^2}}dV},$ where $E$ is bounded by the spheres $4x^2+4y^2+4z^2=1$ and $x^2+y^2+z^2=1.$
2. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.

Express the volume of the solid inside the sphere $x^2+y^2+z^2=16$ and outside the cylinder $x^2+y^2=4$ as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

Express the volume of the solid inside the sphere $x^2+y^2+z^2=16$ and outside the cylinder $x^2+y^2=4$ that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

The power emitted by an antenna has a power density per unit volume given in spherical coordinates by

$p\left(\rho ,\theta ,\phi \right)=\frac{P_0}{{\rho }^2}{\text{cos}}^2\theta {\text{sin}}^4\phi ,$

where $P_0$ is a constant with units in watts. The total power within a sphere $B$ of radius $r$ meters is defined as $P={\displaystyle \underset{B}{\iiint }p\left(\rho ,\theta ,\phi \right)}dV.$ Find the total power $P.$

Use the preceding exercise to find the total power within a sphere $B$ of radius 5 meters when the power density per unit volume is given by $p\left(\rho ,\theta ,\phi \right)=\frac{30}{{\rho }^2}{\text{cos}}^2\theta {\text{sin}}^4\phi .$

A charge cloud contained in a sphere $B$ of radius r centimeters centered at the origin has its charge density given by $q\left(x,y,z\right)=k\sqrt{x^2+y^2+z^2}\frac{\mu \text{C}}{{\text{cm}}^3},$ where $k>0.$ The total charge contained in $B$ is given by $Q={\displaystyle \underset{B}{\iiint }q\left(x,y,z\right)}dV.$ Find the total charge $Q.$

Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is $q\left(x,y,z\right)=20\sqrt{x^2+y^2+z^2}\frac{\mu \text{C}}{{\text{cm}}^3}.$