# 5.4 Triple integrals  (Page 7/8)

 Page 7 / 8

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 $E=\left[0,1\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[0,1\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[0,1\right]$ 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 $x={\text{sin}}^{-1}y,y=0,$ and $x=\frac{\pi }{2}.$

Consider the pyramid with the base in the $xy$ -plane of $\left[-2,2\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[-2,2\right]$ and the vertex at the point $\left(0,0,8\right).$

1. Show that the equations of the planes of the lateral faces of the pyramid are $4y+z=8,$ $4y-z=-8,$ $4x+z=8,$ and $-4x+z=8.$
2. Find the volume of the pyramid.

a. Answers may vary; b. $\frac{128}{3}$

Consider the pyramid with the base in the $xy$ -plane of $\left[-3,3\right]\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left[-3,3\right]$ and the vertex at the point $\left(0,0,9\right).$

1. Show that the equations of the planes of the side faces of the pyramid are $3y+z=9,$ $3y+z=9,$ $y=0$ and $x=0.$
2. Find the volume of the pyramid.

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.

1. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z,$ then with $y,$ and then with $x.$
2. Rewrite the integral in part a. as an equivalent integral in five other orders.

a. $\underset{0}{\overset{4}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{y}^{2}}}{\int }}dz\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx;$ b. $\underset{0}{\overset{2}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{y}^{2}}}{\int }}dz\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{z}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{z}^{2}}}{\int }}dy\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dz,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{x}^{2}-{z}^{2}}}{\int }}dy\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dx,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{z}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}-{z}^{2}}}{\int }}dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz,$ $\underset{0}{\overset{r}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}}}{\int }}\phantom{\rule{0.2em}{0ex}}\underset{0}{\overset{\sqrt{{r}^{2}-{y}^{2}-{z}^{2}}}{\int }}dx\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dy$

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.

1. Write the triple integral that gives the volume of $E$ by integrating first with respect to $z,$ then with $y,$ and then with $x.$
2. Rewrite the integral in part a. as an equivalent integral in five other orders.

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 $\sum _{i=1}^{l}\sum _{j=1}^{m}\sum _{k=1}^{n}f\left(\stackrel{–}{{x}_{i}},\stackrel{–}{{y}_{j}},\stackrel{–}{{z}_{k}}\right)\text{Δ}V,$ where $\left(\stackrel{–}{{x}_{i}},\stackrel{–}{{y}_{j}},\stackrel{–}{{z}_{k}}\right)$ is the center of the box ${B}_{ijk}$ and $\text{Δ}V$ 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]

1. Apply the midpoint rule to approximate $\underset{B}{\iiint }{e}^{\text{−}{x}^{2}}dV$ 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.
2. Use a CAS to improve the above integral approximation in the case of a partition of ${n}^{3}$ cubes of equal size, where $n=3,4\text{,…,}\phantom{\rule{0.2em}{0ex}}10.$

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 $\left(x,y,z\right)$ of a solid $E$ is given by the function $\rho \left(x,y,z\right),$ then the total charge inside the solid is defined as the triple integral $\underset{E}{\iiint }\rho \left(x,y,z\right)dV.$ 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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