5.7 Change of variables in multiple integrals  (Page 8/13)

${\displaystyle \underset{R}{\iint }{e}^{x+y}dA}$

${\displaystyle \underset{R}{\iint }\text{sin}\left(x-y\right)}dA$

${\displaystyle \underset{R}{\iint }\sqrt{x^2+25y^2}dA}$

${{\displaystyle \underset{R}{\iint }\left(x^2+25y^2\right)}}^{2}dA$

${\displaystyle \underset{R}{\iint }\left(x^2-2xy+y^2\right)e^{x+y}dA}$

${\displaystyle \underset{R}{\iint }\left(x^3+3x^{2}y+3x{y}^2+y^3\right)}dA$

The circular annulus sector $R$ bounded by the circles $4x^2+4y^2=1$ and $9x^2+9y^2=64,$ the line $x=y\sqrt{3},$ and the $y\text{-axis}$ is shown in the following figure. Find a transformation $T$ from a rectangular region $S$ in the $r\theta \text{-plane}$ to the region $R$ in the $xy\text{-plane.}$ Graph $S.$

The solid $R$ bounded by the circular cylinder $x^2+y^2=9$ and the planes $z=0,z=1,$ $x=0,\text{and}y=0$ is shown in the following figure. Find a transformation $T$ from a cylindrical box $S$ in $r\theta z\text{-space}$ to the solid $R$ in $xyz\text{-space}.$

Show that ${\displaystyle \underset{R}{\iint }f\left(\sqrt{\frac{x^2}{3}+\frac{y^2}{3}}\right)}dA=2\pi \sqrt{15}{\displaystyle \underset{0}{\overset{1}{\int }}f\left(\rho \right)}\rho d\rho ,$ where $f$ is a continuous function on $\left[0,1\right]$ and $R$ is the region bounded by the ellipse $5x^2+3y^2=15.$

Show that ${\displaystyle \underset{R}{\iiint }f\left(\sqrt{16x^2+4y^2+z^2}\right)}dV=\frac{\pi }{2}{\displaystyle \underset{0}{\overset{1}{\int }}f\left(\rho \right)}{\rho }^{2}d\rho ,$ where $f$ is a continuous function on $\left[0,1\right]$ and $R$ is the region bounded by the ellipsoid $16x^2+4y^2+z^2=1.$

[T] Find the area of the region bounded by the curves $xy=1,xy=3,y=2x,$ and $y=3x$ by using the transformation $u=xy$ and $v=\frac{y}{x}.$ Use a computer algebra system (CAS) to graph the boundary curves of the region $R.$

[T] Find the area of the region bounded by the curves $x^{2}y=2,x^{2}y=3,y=x,$ and $y=2x$ by using the transformation $u=x^{2}y$ and $v=\frac{y}{x}.$ Use a CAS to graph the boundary curves of the region $R.$

Evaluate the triple integral ${\displaystyle \underset{0}{\overset{1}{\int }}{\displaystyle \underset{1}{\overset{2}{\int }}{\displaystyle \underset{z}{\overset{z+1}{\int }}\left(y+1\right)dxdydz}}}$ by using the transformation $u=x-z,$ $v=3y,\text{and}w=\frac{z}{2}.$

Evaluate the triple integral ${\displaystyle \underset{0}{\overset{2}{\int }}{\displaystyle \underset{4}{\overset{6}{\int }}{\displaystyle \underset{3z}{\overset{3z+2}{\int }}\left(5-4y\right)dxdzdy}}}$ by using the transformation $u=x-3z,v=4y,\text{and}w=z.$

A transformation $T:{\text{R}}^2\to {\text{R}}^2,T\left(u,v\right)=\left(x,y\right)$ of the form $x=au+bv,y=cu+dv,$ where $a,b,c,\text{and}d$ are real numbers, is called linear. Show that a linear transformation for which $ad-bc\ne 0$ maps parallelograms to parallelograms.

The transformation $T_{\theta }:{\text{R}}^2\to {\text{R}}^2,T_{\theta }\left(u,v\right)=\left(x,y\right),$ where $x=u\text{cos}\theta -v\text{sin}\theta ,$ $y=u\text{sin}\theta +v\text{cos}\theta ,$ is called a rotation of angle $\theta .$ Show that the inverse transformation of $T_{\theta }$ satisfies $T_{\theta }{}^{\mathrm{-1}}=T_{\text{−}\theta },$ where $T_{\text{−}\theta }$ is the rotation of angle $\text{−}\theta .$

[T] Find the region $S$ in the $uv\text{-plane}$ whose image through a rotation of angle $\frac{\pi }{4}$ is the region $R$ enclosed by the ellipse $x^2+4y^2=1.$ Use a CAS to answer the following questions.

1. Graph the region $S.$
2. Evaluate the integral ${\displaystyle \underset{S}{\iint }{e}^{\mathrm{-2}uv}dudv}.$ Round your answer to two decimal places.

[T] The transformations $T_i:{ℝ}^2\to {ℝ}^2,$ $i=1\text{,…,}4,$ defined by $T_1\left(u,v\right)=\left(u,\text{−}v\right),$ $T_2\left(u,v\right)=\left(\text{−}u,v\right),T_3\left(u,v\right)=\left(\text{−}u,\text{−}v\right),$ and $T_4\left(u,v\right)=\left(v,u\right)$ are called reflections about the $x\text{-axis},y\text{-axis},$ origin, and the line $y=x,$ respectively.

1. Find the image of the region $S=\left\{\left(u,v\right)|u^2+v^2-2u-4v+1\le 0\right\}$ in the $xy\text{-plane}$ through the transformation $T_1\circ T_2\circ T_3\circ T_4.$
2. Use a CAS to graph $R.$
3. Evaluate the integral ${\displaystyle \underset{S}{\iint }\text{sin}(u^2)dudv}$ by using a CAS. Round your answer to two decimal places.

[T] The transformation $T_{k,1,1}:{ℝ}^3\to {ℝ}^3,T_{k,1,1}\left(u,v,w\right)=\left(x,y,z\right)$ of the form $x=ku,$ $y=v,z=w,$ where $k\ne 1$ is a positive real number, is called a stretch if $k>1$ and a compression if $0<k<1$ in the $x\text{-direction}\text{.}$ Use a CAS to evaluate the integral ${\displaystyle \underset{S}{\iiint }{e}^{\text{−}\left(4x^2+9y^2+25z^2\right)}}dxdydz$ on the solid $S=\left\{\left(x,y,z\right)|4x^2+9y^2+25z^2\le 1\right\}$ by considering the compression $T_{2,3,5}\left(u,v,w\right)=\left(x,y,z\right)$ defined by $x=\frac{u}{2},y=\frac{v}{3},$ and $z=\frac{w}{5}.$ Round your answer to four decimal places.