Review Exercises

${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)dydx}=}{\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)dydx}}$

The integral ${\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{r}{\overset{1}{\int }}dzdrd\theta }}}$ represents the volume of a right cone.

${\displaystyle \underset{R}{\iint }\left(5x^3{y}^2-y^2\right)}dA,R=\left\{\left(x,y\right)|0\le x\le 2,1\le y\le 4\right\}$

${\displaystyle \underset{D}{\iint }\text{sin}\left(x^2+y^2\right)}dA$ where $D$ is a disk of radius $2$ centered at the origin

${\displaystyle \underset{\mathrm{-1}}{\overset{1}{\int }}{\displaystyle \underset{0}{\overset{z}{\int }}{\displaystyle \underset{0}{\overset{x-z}{\int }}6dydxdz}}}$

${\displaystyle \underset{0}{\overset{2}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{r}{\overset{1}{\int }}rdzd\theta dr}}}$

${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{\text{−}\sqrt{1-x^2}}{\overset{\sqrt{1-x^2}}{\int }}{\displaystyle \underset{\text{−}\sqrt{1-x^2-y^2}}{\overset{\sqrt{1-x^2-y^2}}{\int }}dz}dydx}}$

The volume of the solid that lies between the paraboloid $z=2x^2+2y^2$ and the plane $z=8.$

The volume of the intersection between two spheres of radius 1, the top whose center is $(0,0,0.25)$ and the bottom, which is centered at $(0,0,0).$

$\rho (x,y)=(y+1)\sqrt{x}$ in the region bounded by $y=e^x,$ $y=0,$ and $x=1.$

The volume an ice cream cone that is given by the solid above $z=\sqrt{\left(x^2+y^2\right)}$ and below $z^2+x^2+y^2=z.$

In reality, it is very likely that the trash at the bottom of Mount Holly has become more compacted with all the weight of the above trash. Consider a density function with respect to height: the density at the top of the mountain is still density $400{\text{lb/ft}}^3$ and the density increases. Every $100$ feet deeper, the density doubles. What is the total weight of Mount Holly?

[T] The density of Earth’s layers is displayed in the table below. Using your calculator or a computer program, find the best-fit quadratic equation to the density. Using this equation, find the total mass of Earth.

Find the volume when you revolve the region around the $y\text{-axis.}$