삼중적분

In the following exercises, evaluate the triple integrals over the bounded region $E$ of the form $E=\{(x,y,z)|{g}_{1}(y)\le x\le {g}_{2}(y),c\le y\le d,e\le z\le f\}.$

$\iiint_{E}(x-yz)dV,$ where $E=\{(x,y,z)|-{y}^{6}\le x\le \sqrt{y},0\le y\le 1,-1\le z\le 1\}$

In the following exercises, evaluate the triple integrals over the bounded region $E$ of the form $E=\{(x,y,z)|{g}_{1}(y)\le x\le {g}_{2}(y),c\le y\le d,e\le z\le f\}.$

$\iiint_{E}zdV,$ where $E=\{(x,y,z)|2-2y\le x\le 2+\sqrt{y},0\le y\le 1,2\le z\le 3\}$

In the following exercises, evaluate the triple integrals over the bounded region

$$E=\{(x,y,z)|{g}_{1}(y)\le x\le {g}_{2}(y),c\le y\le d,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\}.$$

$\iiint_{E}zdV,$ where $E=\{(x,y,z)|-y\le x\le y,0\le y\le 1,0\le z\le 1-{x}^{4}-{y}^{4}\}$

In the following exercises, evaluate the triple integrals over the bounded region

$$E=\{(x,y,z)|{g}_{1}(y)\le x\le {g}_{2}(y),c\le y\le d,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\}.$$

$\iiint_{E}(xz+1)dV,$ where $E=\{(x,y,z)|0\le x\le \sqrt{y},0\le y\le 2,0\le z\le 1-{x}^{2}-{y}^{2}\}$

In the following exercises, evaluate the triple integrals over the bounded region

$$E=\{(x,y,z)|{g}_{1}(y)\le x\le {g}_{2}(y),c\le y\le d,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\}.$$

$\iiint_{E}(x-z)dV,$ where $E=\{(x,y,z)|-\sqrt{1-{y}^{2}}\le x\le 0,0\le y\le \frac{1}{2},0\le z\le 1-{x}^{2}-{y}^{2}\}$

In the following exercises, evaluate the triple integrals over the bounded region

$$E=\{(x,y,z)|{g}_{1}(y)\le x\le {g}_{2}(y),c\le y\le d,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\}.$$

$\iiint_{E}(x+y)dV,$ where $E=\{(x,y,z)|0\le x\le \sqrt{1-{y}^{2}},0\le y\le 1,0\le z\le 1-x\}$

In the following exercises, evaluate the triple integrals over the bounded region

$E=\{(x,y,z)|(x,y)\in D,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\},$ where $D$ is the projection of $E$ onto the $xy$-plane.

$\iint_{D}\left(\int_{1}^{2}(x+z)dz\right)dA,$ where $D=\{(x,y)|{x}^{2}+{y}^{2}\le 1\}$

In the following exercises, evaluate the triple integrals over the bounded region

$E=\{(x,y,z)|(x,y)\in D,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\},$ where $D$ is the projection of $E$ onto the $xy$-plane.

$\iint_{D}\left(\int_{1}^{3}x(z+1)dz\right)dA,$ where $D=\{(x,y)|{x}^{2}-{y}^{2}\ge 1,1\le x\le \sqrt{5}\}$

In the following exercises, evaluate the triple integrals over the bounded region

$E=\{(x,y,z)|(x,y)\in D,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\},$ where $D$ is the projection of $E$ onto the $xy$-plane.

$\iint_{D}\left(\int_{0}^{10-x-y}(x+2z)dz\right)dA,$ where $D=\{(x,y)|y\ge 0,x\ge 0,x+y\le 10\}$

In the following exercises, evaluate the triple integrals over the bounded region

$E=\{(x,y,z)|(x,y)\in D,{u}_{1}(x,y)\le z\le {u}_{2}(x,y)\},$ where $D$ is the projection of $E$ onto the $xy$-plane.

$\iint_{D}\left(\int_{0}^{4{x}^{2}+4{y}^{2}}y\,dz\right)dA,$ where $D=\{(x,y)|{x}^{2}+{y}^{2}\le 4,y\ge 1,x\ge 0\}$

The solid $E$ bounded by ${y}^{2}+{z}^{2}=9,z=0,x=0,$ and $x=5$ is shown in the following figure. Evaluate the integral $\iiint_{E}z\,dV$ by integrating first with respect to $z,$ then $y,\,\text{and then}\,x.$

The solid $E$ bounded by $y=\sqrt{x},$ $x=4,$ $y=0,$ $z=-2$, and $z=1$ is given in the following figure. Evaluate the integral $\iiint_{E}xyz\,dV$ by integrating first with respect to $x,$ then $y,$ and then $z.$

[T] The volume of a solid $E$ is given by the integral $\int_{-2}^{0}\,\int_{x}^{0}\,\int_{0}^{{x}^{2}+{y}^{2}}dz\,dy\,dx.$ Use a computer algebra system (CAS) to graph $E$ and find its volume. Round your answer to two decimal places.

[T] The volume of a solid $E$ is given by the integral $\int_{-1}^{0}\,\int_{-{x}^{2}}^{0}\,\int_{0}^{1+\sqrt{{x}^{2}+{y}^{2}}}dz\,dy\,dx.$ Use a CAS to graph $E$ and find its volume $V.$ Round your answer to two decimal places.

In the following exercises, use two circular permutations of the variables $x,y,\,\text{and}\,z$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x,y,\,\text{and}\,z$ is the arrangement of the numbers in one of the following orders: $y,z,\,\text{and}\,x\,\text{or}\,z,x,\,\text{and}\,y.$

$\int_{0}^{1}\,\int_{1}^{3}\,\int_{2}^{4}({x}^{2}{z}^{2}+1)dx\,dy\,dz$

In the following exercises, use two circular permutations of the variables $x,y,\,\text{and}\,z$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x,y,\,\text{and}\,z$ is the arrangement of the numbers in one of the following orders: $y,z,\,\text{and}\,x\,\text{or}\,z,x,\,\text{and}\,y.$

$\int_{1}^{3}\,\int_{0}^{1}\,\int_{0}^{\text{−y}+1}(2y+5z+7x)dz\,dy\,dx$

In the following exercises, use two circular permutations of the variables $x,y,\,\text{and}\,z$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x,y,\,\text{and}\,z$ is the arrangement of the numbers in one of the following orders: $y,z,\,\text{and}\,x\,\text{or}\,z,x,\,\text{and}\,y.$

$\int_{0}^{1}\,\int_{-y}^{y}\,\int_{0}^{1-{x}^{4}-{y}^{4}}{e}^{x}\,dz\,dx\,dy$

In the following exercises, use two circular permutations of the variables $x,y,\,\text{and}\,z$ to write new integrals whose values equal the value of the original integral. A circular permutation of $x,y,\,\text{and}\,z$ is the arrangement of the numbers in one of the following orders: $y,z,\,\text{and}\,x\,\text{or}\,z,x,\,\text{and}\,y.$

$\int_{-1}^{1}\,\int_{0}^{1}\,\int_{-{y}^{6}}^{\sqrt{y}}(x+yz)dx\,dy\,dz$

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 $a>0.$