삼중적분

Set up the integral that gives the volume of the solid $E$ bounded by ${y}^{2}={x}^{2}+{z}^{2}$ and $y=a,$ where $a>0$ and $y \ge 0$

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 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 $x={\sin }^{-1}y,y=0,$ and $x=\frac{\pi }{2}.$

Find the volume of the prism with vertices $(0,0,0),(2,0,0),(2,3,0),$ $(0,3,0),(0,0,1),\,\text{and}\,(2,0,1).$

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

The midpoint rule for the triple integral $\iiint_{B}f(x,y,z)dV$ 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 $\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}f\left(\overset{-}{{x}_{i}},\overset{-}{{y}_{j}},\overset{-}{{z}_{k}}\right)\Delta V,$ where $\left(\overset{-}{{x}_{i}},\overset{-}{{y}_{j}},\overset{-}{{z}_{k}}\right)$ is the center of the box ${B}_{ijk}$ and $\Delta V$ is the volume of each subbox. Apply the midpoint rule to approximate $\iiint_{B}{x}^{2}dV$ over the solid $B=\{(x,y,z)|0\le x\le 1,0\le y\le 1,0\le z\le 1\}$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

Suppose that the temperature in degrees Celsius at a point $(x,y,z)$ of a solid $E$ bounded by the coordinate planes and $x+y+z=5$ is $T(x,y,z)=xz+5z+10.$ Find the average temperature over the solid.

Suppose that the temperature in degrees Fahrenheit at a point $(x,y,z)$ of a solid $E$ bounded by the coordinate planes and $x+y+z=5$ is $T(x,y,z)=x+y+xy.$ Find the average temperature over the solid.