5.4 Triple integrals  (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 $z=5.$

Find the average value of the function $f\left(x,y,z\right)=xyz$ over the solid $E=\left[0,1\right]\times \left[0,1\right]\times \left[0,1\right]$ 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 $x+y=7.$

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 $\left[\mathrm{-2},2\right]\times \left[\mathrm{-2},2\right]$ 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 $4y+z=8,$ $4y-z=\mathrm{-8},$ $4x+z=8,$ and $\mathrm{-4}x+z=8.$
2. Find the volume of the pyramid.

Consider the pyramid with the base in the $xy$ -plane of $\left[\mathrm{-3},3\right]\times \left[\mathrm{-3},3\right]$ 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.

The solid $E$ bounded by the sphere of equation $9x^2+4y^2+z^2=1$ 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 $\left(0,0,0\right),\left(2,0,0\right),\left(2,3,0\right),$ $\left(0,3,0\right),\left(0,0,1\right),\text{and}\left(2,0,1\right).$

Find the volume of the prism with vertices $\left(0,0,0\right),\left(4,0,0\right),\left(4,6,0\right),$ $\left(0,6,0\right),\left(0,0,1\right),\text{and}\left(4,0,1\right).$

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.

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 ${\displaystyle \underset{B}{\iiint }f\left(x,y,z\right)dV}$ 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 ${\displaystyle \sum _{i=1}^l{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{k=1}^{n}f\left(\overline{x_i},\overline{y_j},\overline{z_k}\right)}}}\text{Δ}V,$ where $\left(\overline{x_i},\overline{y_j},\overline{z_k}\right)$ is the center of the box $B_{ijk}$ and $\text{Δ}V$ is the volume of each subbox. Apply the midpoint rule to approximate ${\displaystyle \underset{B}{\iiint }{x}^{2}dV}$ 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.

[T]

1. Apply the midpoint rule to approximate ${\displaystyle \underset{B}{\iiint }{e}^{\text{−}{x}^2}dV}$ over the solid $B=\left\{(x,y,z)|0\le x\le 1,0\le y\le 1,0\le z\le 1\right\}$ 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 $n=3,4\text{,…,}10.$

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.

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 $\frac{3a^3\sqrt{3}}{2}$ 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 ${\displaystyle \underset{E}{\iiint }\rho (x,y,z)dV}.$ 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 $E.$