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Set up the integral that gives the volume of the solid $E$ bounded by $x={y}^{2}+{z}^{2}$ and $x={a}^{2},$ where $a>0.$
Find the average value of the function $f\left(x,y,z\right)=x+y+z$ over the parallelepiped determined by $x=0,x=1,y=0,y=3,z=0,$ and $z=5.$
$\frac{9}{2}$
Find the average value of the function $f\left(x,y,z\right)=xyz$ over the solid $E=\left[0,1\right]\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left[0,1\right]\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left[0,1\right]$ situated in the first octant.
Find the volume of the solid $E$ that lies under the plane $x+y+z=9$ and whose projection onto the $xy$ -plane is bounded by $x=\sqrt{y-1},x=0,$ and $x+y=7.$
$\frac{156}{5}$
Find the volume of the solid E that lies under the plane $2x+y+z=8$ and whose projection onto the $xy$ -plane is bounded by $x={\text{sin}}^{\mathrm{-1}}y,y=0,$ and $x=\frac{\pi}{2}.$
Consider the pyramid with the base in the $xy$ -plane of $\left[\mathrm{-2},2\right]\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left[\mathrm{-2},2\right]$ and the vertex at the point $\left(0,0,8\right).$
a. Answers may vary; b. $\frac{128}{3}$
Consider the pyramid with the base in the $xy$ -plane of $\left[\mathrm{-3},3\right]\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left[\mathrm{-3},3\right]$ and the vertex at the point $\left(0,0,9\right).$
The solid $E$ bounded by the sphere of equation ${x}^{2}+{y}^{2}+{z}^{2}={r}^{2}$ with $r>0$ and located in the first octant is represented in the following figure.
a. $\underset{0}{\overset{4}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}}}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{y}^{2}}}{\int}}dz\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx}}};$ b. $\underset{0}{\overset{2}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}}}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{y}^{2}}}{\int}}dz\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy}}},$ $\underset{0}{\overset{r}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{z}^{2}}}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{z}^{2}}}{\int}}dy\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dz}}},$ $\underset{0}{\overset{r}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}}}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{z}^{2}}}{\int}}dy\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dx}}},$ $\underset{0}{\overset{r}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{z}^{2}}}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}-{z}^{2}}}{\int}}dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz}}},$ $\underset{0}{\overset{r}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}}}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}-{z}^{2}}}{\int}}dx\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dy}}$
The solid $E$ bounded by the sphere of equation $9{x}^{2}+4{y}^{2}+{z}^{2}=1$ and located in the first octant is represented in the following figure.
Find the volume of the prism with vertices $\left(0,0,0\right),\left(2,0,0\right),\left(2,3,0\right),$ $\left(0,3,0\right),\left(0,0,1\right),\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(2,0,1\right).$
$3$
Find the volume of the prism with vertices $\left(0,0,0\right),\left(4,0,0\right),\left(4,6,0\right),$ $\left(0,6,0\right),\left(0,0,1\right),\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(4,0,1\right).$
The solid $E$ bounded by $z=10-2x-y$ and situated in the first octant is given in the following figure. Find the volume of the solid.
$\frac{250}{3}$
The solid $E$ bounded by $z=1-{x}^{2}$ and situated in the first octant is given in the following figure. Find the volume of the solid.
The midpoint rule for the triple integral $\underset{B}{\iiint}f\left(x,y,z\right)dV$ over the rectangular solid box $B$ is a generalization of the midpoint rule for double integrals. The region $B$ is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum $\sum _{i=1}^{l}{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{k=1}^{n}f\left(\stackrel{\u2013}{{x}_{i}},\stackrel{\u2013}{{y}_{j}},\stackrel{\u2013}{{z}_{k}}\right)}}}\text{\Delta}V,$ where $\left(\stackrel{\u2013}{{x}_{i}},\stackrel{\u2013}{{y}_{j}},\stackrel{\u2013}{{z}_{k}}\right)$ is the center of the box ${B}_{ijk}$ and $\text{\Delta}V$ is the volume of each subbox. Apply the midpoint rule to approximate $\underset{B}{\iiint}{x}^{2}dV$ over the solid $B=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le 1,0\le z\le 1\right\}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.
$\frac{5}{16}\approx 0.313$
[T]
Suppose that the temperature in degrees Celsius at a point $\left(x,y,z\right)$ of a solid $E$ bounded by the coordinate planes and $x+y+z=5$ is $T\left(x,y,z\right)=xz+5z+10.$ Find the average temperature over the solid.
$\frac{35}{2}$
Suppose that the temperature in degrees Fahrenheit at a point $\left(x,y,z\right)$ of a solid $E$ bounded by the coordinate planes and $x+y+z=5$ is $T\left(x,y,z\right)=x+y+xy.$ Find the average temperature over the solid.
Show that the volume of a right square pyramid of height $h$ and side length $a$ is $v=\frac{h{a}^{2}}{3}$ by using triple integrals.
Show that the volume of a regular right hexagonal prism of edge length $a$ is $\frac{3{a}^{3}\sqrt{3}}{2}$ by using triple integrals.
Show that the volume of a regular right hexagonal pyramid of edge length $a$ is $\frac{{a}^{3}\sqrt{3}}{2}$ by using triple integrals.
If the charge density at an arbitrary point $(x,y,z)$ of a solid $E$ is given by the function $\rho (x,y,z),$ then the total charge inside the solid is defined as the triple integral $\underset{E}{\iiint}\rho (x,y,z)dV}.$ Assume that the charge density of the solid $E$ enclosed by the paraboloids $x=5-{y}^{2}-{z}^{2}$ and $x={y}^{2}+{z}^{2}-5$ is equal to the distance from an arbitrary point of $E$ to the origin. Set up the integral that gives the total charge inside the solid $E.$
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