Change of variables

A change of variables replaces $x$ and $y$ with new variables $u$ and $v$ using equations $x=x(u,v)$ and $y=y(u,v)$. The area element transforms as $dxdy=\left|\frac{\partial (x,y)}{\partial (u,v)}\right|dudv$, where the Jacobian determinant is $\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cc}{x}_u& x_v\\ y_u& y_v\end{array}\right|=x_u{y}_v-x_v{y}_u$. In polar coordinates, $x=r\cos \theta$, $y=r\sin \theta$, so the Jacobian is $r$ and $dxdy=rdrd\theta$. To integrate over a region, find the new bounds for $u$ and $v$, insert the absolute value of the Jacobian, and evaluate the iterated integral.