For a change of variables from $x$ and $y$ to $u$ and $v$ given by $x=g_{1}(u,v)$ and $y=g_{2}(u,v),$ the Jacobian is the determinant _____.

True or False When introducing a Jacobian into an integral because of a change of variables, the absolute value of the Jacobian is used.

$x=u+v,\quad y=u-v$

$x=u+5,\quad y=v-7$

$x=4u-v,\quad y=3u+\dfrac{1}{2}v$

$x=u^{2}-v^{2},\quad y=u^{2}+v^{2}$

$x=\dfrac{u}{v},\quad y=u+v$

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

$x=ve^{u},\quad y=ue^{-v}$

$x=u\cos v,\quad y=v\sin u$

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

$x=2u+v+w,\quad y=u+v-w,\quad z=u-v$

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

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

$x=u,\quad y=v^{2},\quad z=w^{3}$

$x=3 ( u+v) ,\quad y=u-w,\quad z=u^{2}-w^{2}$

Area Find the area of the region $R$ enclosed by $y=4\sqrt{1- \dfrac{x^{2}}{9}}$ and $y=0,$ using the change of variables $u=\dfrac{x}{3}$ and $v=\dfrac{y}{2}.$

Area Find the area of the region $R$ enclosed by $xy=1,$ $xy=3,$ $y=x,$ and $y=3x$ using the change of variables $u=x$ and $v=xy.$ See the figure.

$\iint\limits_{\kern-3ptR}x\,{\it dA}$, where $R$ is the region enclosed by $2x-y=0$, $2x-y=4$, $x+y=0$, and $x+y=3$. Use the change of variables $u=2x-y$ and $v=x+y$.

$\iint\limits_{\kern-3ptR}(x^{2}+y^{2})\,{\it dA}$, where $R$ is the region enclosed by $2x-y=1$, $2x-y=3$, $x+y=1$, and $x+y=2$. Use the change of variables $u=2x-y$ and $v=x+y.$

$\iint\limits_{\kern-3ptR}xy\,{\it dA},$ where $R$ is the triangular region whose vertices are $(-1,1)$, $(1,1)$, and $(0,0)$. Use the change of variables $u=x+y$ and $v=x-y$.

$\iint\limits_{\kern-3ptR}(x+ y)$ sin$(x-y) {\it dA}$, where $R$ is the triangular region whose vertices are $(-1,1)$, $(1,1)$, and $(0,0)$; use the change of variables $u=x+y$ and $v=x-y$.

$\iint\limits_{\kern-3ptR}(3x-2y)\,{\it dA}$, where $R$ is the region enclosed by the ellipse $9x^{2}+16y^{2}=144.$ Change the variables using $u=\dfrac{x}{4}$ and $v=\dfrac{y}{3}.$

$\iint\limits_{\kern-3ptR}(x^{2}-y)$, ${\it dA}$, where $R$ is the region enclosed by the ellipse $9x^{2}+16y^{2}=144.$ Change the variables using $u=\dfrac{x}{4}$ and $v=\dfrac{y}{3}.$

$\iiint\limits_{\kern-3ptE}(x+1)\,{\it dV}$, where $E$ is the solid enclosed by the ellipsoid $\dfrac{x^{2}}{4}+\dfrac{y^{2}}{4}+ \dfrac{z^{2}}{9}=1.$ Change the variables using $u=\dfrac{x}{2}$, $v=\dfrac{y }{2}$, and $w=\dfrac{z}{3}.$

$\iiint\limits_{\kern-3ptE}(x+2y)\,{\it dV}$, where $E$ is the region enclosed by the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}+z^{2}=1.$ Change the variables using $u= \dfrac{x}{3}$, $v=\dfrac{y}{2},$ and $w=z$.

$\iint\limits_{\kern-3ptR}x^{2}y\,{\it dA}$, where $R$ is the region enclosed by the lines $2x-3y=0$, $2x-3y=1$, $x+2y=0$, and $x+2y=3$

$\iint\limits_{\kern-3ptR}x\,{\it dA}$, where $R$ is the region enclosed by the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{16}=1$

$\iint\limits_{\kern-3ptR}x\cos (xy) \,{\it dA}$, where $R$ is the region enclosed by $xy=1$, $xy=3$, $x=1$, and $x=3$

$\iint\limits_{\kern-3ptR}ye^{xy}\,{\it dA}$, where $R$ is the region enclosed by $xy=1$, $xy=3$, $x=1$, and $x=3$

Area Use a change of variables to find the area of the region $R$ enclosed by the lines $y=3x,$ $y-3x=2,$ $2y+x=0$, and $y=-\dfrac{1}{2} x+5$.

Area Use a change of variables to find the area of the region $R$ enclosed by the lines $x+y=1$, $x+y=-3,$ $y=2x$, and $y=2x+4$.

Volume Use a change of variables to find the volume $V$ of the solid $E$ enclosed by the planes $x+2y=0,$ $x+2y=3,$ $y-z=0,$ $y-z=2$, $z=0,$ and $z=6.$

Volume Use a change of variables to find the volume $V$ of the solid $E$ enclosed by the planes $x-y=0,$ $x-y=3,$ $z-2x=0,$ $z-2x=4$, $z=1,$ and $z=5.$

Volume Find the volume of the solid $E$ enclosed by the paraboloid $z=9-$ $x^{2}-9y^{2}$ and the plane $z=1,$ given the change of variables $x=3u\cos v,$ $y=u\sin v,$ and $z=w$. See the figure.

Volume Find the volume of the solid $E$ enclosed by the ellipsoid $\dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}+z^{2}=1$ above the $xy$-plane, given the change of variables $u=\dfrac{x}{2}$, $v=\dfrac{y}{3},$ and $w=z.$

Show that in changing from rectangular coordinates $(x, y, z)$ to spherical coordinates $(\rho, \theta, \phi)$, the triple integral of $f$ over $E$ becomes $$\begin{eqnarray*} &&\iiint\limits_{\kern-18ptE}f(x, y, z)\,{\it dV}\\[4pt] &&\quad=\iiint\limits_{\kern-18ptE^{\#}}f(\rho \sin \phi \cos \theta , \rho \sin \phi \sin \theta , \rho \cos \phi )\rho ^{2}\sin \phi \,d\rho \,d\theta \,d\phi \end{eqnarray*}$$

A region $R$ is enclosed by the graphs of $xy=2,$ $xy=5,$ $y= \sqrt{x},$ and $y=3\sqrt{x}$.