# Need help with double integrals and change of variables.

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?

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Let $u = x^2 - y^2$ $v = x^2 + y^2$
When dealing with problems like this, a really good question to ask yourself is, "how can I make my limits of integration easier?" For instance, sometimes converting to polar makes the limits really easy, but keep your mind open to new substitutions, especially when you don't have to integrate from zero to r or some other variable or function. The best possible case is when you can get both of your regions in terms of real values, as we did by letting $u = x^2 - y^2$ $v = x^2 + y^2$ Now the only tricky part is the Jacobian. If you want a hint with that, let me know.