다중적분 전체 문제

In the following exercises, find the Jacobian $J$ of the transformation.

$x=u{e}^{v},y={e}^{-v}$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=u\,\cos ({e}^{v}),y=u\,\sin ({e}^{v})$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=v\,\sin ({u}^{2}),y=v\,\cos ({u}^{2})$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=u\,\cosh \,v,y=u\,\sinh \,v,z=w$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=v\,\cosh \left(\frac{1}{u}\right),y=v\,\sinh \left(\frac{1}{u}\right),z=u+{w}^{2}$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=u+v,y=v+w,z=u$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=u-v,y=u+v,z=u+v+w$

The triangular region $R$ with the vertices $(0,0),(1,1),\text{and}\,(1,2)$ is shown in the following figure.

a. Find a transformation $T:S\to R,$ $T(u,v)=(x,y)=(au+bv,cu+dv),$ where $a,b,c,$ and $d$ are real numbers with $ad-bc\ne 0$ such that ${T}^{-1}(0,0)=(0,0),{T}^{-1}(1,1)=(1,0),$ and ${T}^{-1}(1,2)=(0,1).$ b. Use the transformation $T$ to find the area $A(R)$ of the region $R.$

The triangular region $R$ with the vertices $(0,0),(2,0),\text{and}\,(1,3)$ is shown in the following figure.

a. Find a transformation $T:S\to R,$ $T(u,v)=(x,y)=(au+bv,cu+dv),$ where $a,b,c$ and $d$ are real numbers with $ad-bc\ne 0$ such that ${T}^{-1}(0,0)=(0,0),$ ${T}^{-1}(2,0)=(1,0),$ and ${T}^{-1}(1,3)=(0,1).$ b. Use the transformation $T$ to find the area $A(R)$ of the region $R.$

In the following exercises, use the transformation $u=y-x,v=y,$ to evaluate the integrals on the parallelogram $R$ of vertices $(0,0),(1,0),(2,1),\text{and}\,(1,1)$ shown in the following figure.

$\iint_{R}(y-x)dA$

In the following exercises, use the transformation $u=y-x,v=y,$ to evaluate the integrals on the parallelogram $R$ of vertices $(0,0),(1,0),(2,1),\text{and}\,(1,1)$ shown in the following figure.

$\iint_{R}({y}^{2}-xy)dA$

In the following exercises, use the transformation $y-x=u,x+y=v$ to evaluate the integrals on the square $R$ determined by the lines $y=x,y=-x+2,y=x+2,$ and $y=-x$ shown in the following figure.

$\iint_{R}{e}^{x+y}dA$

In the following exercises, use the transformation $y-x=u,x+y=v$ to evaluate the integrals on the square $R$ determined by the lines $y=x,y=-x+2,y=x+2,$ and $y=-x$ shown in the following figure.

$\iint_{R}\sin (x-y)dA$

In the following exercises, use the transformation $x=u,5y=v$ to evaluate the integrals on the region $R$ bounded by the ellipse ${x}^{2}+25{y}^{2}=1$ shown in the following figure.

$\iint_{R}\sqrt{{x}^{2}+25{y}^{2}}\,dA$

In the following exercises, use the transformation $x=u,5y=v$ to evaluate the integrals on the region $R$ bounded by the ellipse ${x}^{2}+25{y}^{2}=1$ shown in the following figure.

${\iint_{R}({x}^{2}+25{y}^{2})}^{2}dA$

In the following exercises, use the transformation $u=x+y,v=x-y$ to evaluate the integrals on the trapezoidal region $R$ determined by the points $(1,0),(2,0),(0,2),\text{and}\,(0,1)$ shown in the following figure.

$\iint_{R}({x}^{2}-2xy+{y}^{2}){e}^{x+y}dA$

In the following exercises, use the transformation $u=x+y,v=x-y$ to evaluate the integrals on the trapezoidal region $R$ determined by the points $(1,0),(2,0),(0,2),\text{and}\,(0,1)$ shown in the following figure.

$\iint_{R}({x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3})dA$

The circular annulus sector $R$ bounded by the circles $4{x}^{2}+4{y}^{2}=1$ and $9{x}^{2}+9{y}^{2}=64,$ 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 $\iint_{R}f\left(\sqrt{\frac{{x}^{2}}{3}+\frac{{y}^{2}}{3}}\right)dA=2\pi \sqrt{15}\int_{0}^{1}f(\rho )\rho \,d\rho ,$ where $f$ is a continuous function on $[0,1]$ and $R$ is the region bounded by the ellipse $5{x}^{2}+3{y}^{2}=15.$