### Review Exercises

*True or False?* Justify your answer with a proof or a counterexample.

$\underset{a}{\overset{b}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{c}{\overset{d}{\int}}f\left(x,y\right)dy\phantom{\rule{0.2em}{0ex}}dx}=}{\displaystyle \underset{c}{\overset{d}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{a}{\overset{b}{\int}}f\left(x,y\right)dy\phantom{\rule{0.2em}{0ex}}dx}$

Fubini’s theorem can be extended to three dimensions, as long as $f$ is continuous in all variables.

The integral $\underset{0}{\overset{2\pi}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{1}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{r}{\overset{1}{\int}}dz\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\theta}}$ represents the volume of a right cone.

The Jacobian of the transformation $x={u}^{2}-2v,y=3v-2uv$ is given by $\mathrm{-4}{u}^{2}+6u+4v.$

Evaluate the following integrals.

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

$\underset{D}{\iint}\frac{y}{3{x}^{2}+1}}dA,D=\left\{\left(x,y\right)|0\le x\le 1,\text{\u2212}x\le y\le x\right\$

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

$\underset{0}{\overset{1}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{y}{\overset{1}{\int}}xy{e}^{{x}^{2}}dx\phantom{\rule{0.2em}{0ex}}dy}$

$\underset{\mathrm{-1}}{\overset{1}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{z}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{x-z}{\int}}6dy\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dz}}$

$\underset{R}{\iiint}3y\phantom{\rule{0.2em}{0ex}}dV},$ where $R=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le x,0\le z\le \sqrt{9-{y}^{2}}\right\}$

$\underset{0}{\overset{2}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{2\pi}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{r}{\overset{1}{\int}}r\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}d\theta \phantom{\rule{0.2em}{0ex}}dr}}$

$\underset{0}{\overset{2\pi}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{0}{\overset{\pi \text{/}2}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{1}{\overset{3}{\int}}{\rho}^{2}\text{sin}(\phi )d\rho \phantom{\rule{0.2em}{0ex}}d\phi \phantom{\rule{0.2em}{0ex}}d\theta}}$

$\underset{0}{\overset{1}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{\text{\u2212}\sqrt{1-{x}^{2}}}{\overset{\sqrt{1-{x}^{2}}}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{\text{\u2212}\sqrt{1-{x}^{2}-{y}^{2}}}{\overset{\sqrt{1-{x}^{2}-{y}^{2}}}{\int}}dz}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dx}$

For the following problems, find the specified area or volume.

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

The volume of the solid bounded by the cylinder ${x}^{2}+{y}^{2}=16$ and from $z=1$ and $z+x=2.$

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 $(0,0,0).$

For the following problems, find the center of mass of the region.

$\rho (x,y)=(y+1)\sqrt{x}$ 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 ${z}^{2}+{x}^{2}+{y}^{2}=z.$

The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a right circular cone of height $1100$ ft and radius $6000$ ft.

If the compacted trash used to build Mount Holly on average has a density $400{\phantom{\rule{0.2em}{0ex}}\text{lb/ft}}^{3},$ find the amount of work required to build the mountain.

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 $400{\phantom{\rule{0.2em}{0ex}}\text{lb/ft}}^{3}$ and the density increases. Every $100$ feet deeper, the density doubles. What is the total weight of Mount Holly?

The following problems consider the temperature and density of Earth’s layers.

**[T]** The temperature of Earth’s layers is exhibited in the table below. Use your calculator to fit a polynomial of degree $3$ to the temperature along the radius of the Earth. Then find the average temperature of Earth. (*Hint*: begin at $0$ in the inner core and increase outward toward the surface)

Layer | Depth from center (km) | Temperature $\text{\xb0}C$ |
---|---|---|

Rocky Crust | 0 to 40 | 0 |

Upper Mantle | 40 to 150 | 870 |

Mantle | 400 to 650 | 870 |

Inner Mantel | 650 to 2700 | 870 |

Molten Outer Core | 2890 to 5150 | 4300 |

Inner Core | 5150 to 6378 | 7200 |

**[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.

Layer | Depth from center (km) | Density (g/cm3) |
---|---|---|

Inner Core | $0$ | $12.95$ |

Outer Core | $1228$ | $11.05$ |

Mantle | $3488$ | $5.00$ |

Upper Mantle | $6338$ | $3.90$ |

Crust | $6378$ | $2.55$ |

The following problems concern the Theorem of Pappus (see Moments and Centers of Mass for a refresher), a method for calculating volume using centroids. Assuming a region $R,$ when you revolve around the $x\text{-axis}$ the volume is given by ${V}_{x}=2\pi A\stackrel{\u2013}{\left|y\right|},$ and when you revolve around the $\text{y-axis}$ the volume is given by ${V}_{y}=2\pi A\left|\stackrel{\u2013}{x}\right|,$ where $A$ is the area of $R.$ Consider the region bounded by ${x}^{2}+{y}^{2}=1$ and above $y=x+1.$

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