5.7 Change of variables in multiple integrals  (Page 9/13)

[T] The transformation $T_{a,0}:{ℝ}^2\to {ℝ}^2,T_{a,0}\left(u,v\right)=\left(u+av,v\right),$ where $a\ne 0$ is a real number, is called a shear in the $x\text{-direction}\text{.}$ The transformation, $T_{b,0}:{\text{R}}^2\to {\text{R}}^2,T_{o,b}\left(u,v\right)=\left(u,bu+v\right),$ where $b\ne 0$ is a real number, is called a shear in the $y\text{-direction}\text{.}$

1. Find transformations $T_{0,2}\circ T_{3,0}.$
2. Find the image $R$ of the trapezoidal region $S$ bounded by $u=0,v=0,v=1,$ and $v=2-u$ through the transformation $T_{0,2}\circ T_{3,0}.$
3. Use a CAS to graph the image $R$ in the $xy\text{-plane}\text{.}$
4. Find the area of the region $R$ by using the area of region $S.$

Use the transformation, $x=au,y=av,z=cw$ and spherical coordinates to show that the volume of a region bounded by the spheroid $\frac{x^2+y^2}{a^2}+\frac{z^2}{c^2}=1$ is $\frac{4\pi a^{2}c}{3}.$

Find the volume of a football whose shape is a spheroid $\frac{x^2+y^2}{a^2}+\frac{z^2}{c^2}=1$ whose length from tip to tip is $11$ inches and circumference at the center is $22$ inches. Round your answer to two decimal places.

[T] Lamé ovals (or superellipses) are plane curves of equations ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=1,$ where a , b , and n are positive real numbers.

1. Use a CAS to graph the regions $R$ bounded by Lamé ovals for $a=1,b=2,n=4$ and $n=6,$ respectively.
2. Find the transformations that map the region $R$ bounded by the Lamé oval $x^4+y^4=1,$ also called a squircle and graphed in the following figure, into the unit disk.
   
3. Use a CAS to find an approximation of the area $A\left(R\right)$ of the region $R$ bounded by $x^4+y^4=1.$ Round your answer to two decimal places.

[T] Lamé ovals have been consistently used by designers and architects. For instance, Gerald Robinson, a Canadian architect, has designed a parking garage in a shopping center in Peterborough, Ontario, in the shape of a superellipse of the equation ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=1$ with $\frac{a}{b}=\frac{9}{7}$ and $n=e.$ Use a CAS to find an approximation of the area of the parking garage in the case $a=900$ yards, $b=700$ yards, and $n=2.72$ yards.

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

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

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.

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

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

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

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

${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{y}{\overset{1}{\int }}xy{e}^{x^2}dxdy}}$

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

${\displaystyle \underset{R}{\iiint }3ydV},$ where $R=\left\{\left(x,y,z\right)|0\le x\le 1,0\le y\le x,0\le z\le \sqrt{9-y^2}\right\}$

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

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

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

The area of region enclosed by one petal of $r=\text{cos}\left(4\theta \right).$

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

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

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 $(0,0,0).$

$\rho (x,y)=xy$ on the circle with radius $1$ in the first quadrant only.

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

$\rho (x,y,z)=z$ on the inverted cone with radius $2$ and height $2.$

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

If the compacted trash used to build Mount Holly on average has a density $400{\text{lb/ft}}^3,$ 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 $400{\text{lb/ft}}^3$ and the density increases. Every $100$ feet deeper, the density doubles. What is the total weight of Mount Holly?

[T] The 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)

[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 $x\text{-axis.}$

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