I'm trying to calculate the integral of $(x^2 + y^2) \;dx\;dy$ where it is bounded by the positive region where the four curves intersect.
$x^2 - y^2 =2$
$x^2 - y^2 =8$
$x^2 + y^2 =8$
$x^2 + y^2 =16$
So I know I need to do a change of variables and polar looks like a good choice to me. This is where I get lost. So I use $x=r\cos t$, $y=r\sin t$ to change to polar, calculate the Jacobian from $x(r,t)$ and $y(r,t)$ Then do the integral $(x(r,t)^2+y(r,t)^2)$ (jacobian) $drdt$ ? And how do I figure out the limits of integration?