Triple Integrals

(requires JavaScript)

1. Integrate ${\displaystyle \underset{B}{\iiint }xy{z}^{2}dV}$ over the box $B=\left[0,1\right]\times \left[-1,2\right]\times \left[0,3\right]$ first with respect to $z$, then $x$, and then $y$.
   $\frac{27}{4}$
2. Evaluate ${\displaystyle {\int }_0^3{\int }_0^1{\int }_0^{\sqrt{1-z^2}}z{e}^{y}dxdzdy}$.
   $\frac{1}{3}\left(e^3-1\right)$

3. Evaluate the triple integral ${\displaystyle \underset{E}{\iiint }yz\cos <!--FUNCTION\; APPLICATION-->\left(x^5\right)dV}$ where

   $E=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le x,x\le z\le 2x\right\}$.

4. Evaluate the triple integral ${\displaystyle \underset{E}{\iiint }ydV}$ where $E$ is bounded by the planes $x=0$, $y=0$, $z=0$, and $2x+2y+z=4$.

   $\frac{4}{3}$

5. Evaluate the triple integral ${\displaystyle \underset{E}{\iiint }xdV}$ where $E$ is bounded by the paraboloid $x=4y^2+4z^2$ and the plane $x=4$.

   ${\displaystyle \frac{16\pi }{3}}$
6. Use a triple integral to find the volume of the solid bounded by the cylinder $x^2+y^2=9$ and the planes $y+z=5$ and $z=1$.
   $36\pi$
7. Express the integral ${\displaystyle \underset{E}{\iiint }f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)dV}$ as an iterated integral in $6$ different ways if $E$ is the solid bounded by the surfaces $z=0$, $x=0$, $y=2$, and $z=y-2x$.

8. The figure shows the region of integration for the integral ${\displaystyle {\int }_0^1{\int }_{\sqrt{x}}^1{\int }_0^{1-y}f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)dzdydx}$. Rewrite this integral as an equivalent iterated integral in the $5$ other orders.

   

9. Write $5$ other iterated integrals that are equal to the integral ${\displaystyle {\int }_0^1{\int }_y^1{\int }_0^{y}f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)dzdxdy}$.
10. Find the average value of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=x^{2}z+y^{2}z$ over the region enclosed by the paraboloid $z=1-x^2-y^2$ and the plane $z=0$.

11. Find the region $E$ for which the triple integral

    ${\displaystyle \underset{E}{\iiint }\left(1-x^2-2y^2-3z^2\right)dV}$

    is a maximum.

    The region bounded by the ellipsoid $x^2+2y^2+3z^2=1$