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$\begingroup$

Where D is the domain given by: $ D:=\{(x,y) : x \geq 0, -1 \leq y \leq 2, 4 \leq x^2 + y ^2 \leq 16 , 1\leq x^2 - y^2 \leq 9\} $

I'm not sure what change of variables to use here.

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    **Hint** Try the change of variables $u=x^2,\ v=y^2$. (Disclaimer: I haven't verified this is helpful.)2012-11-04
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    @pootieman, Are you really sure?2012-11-04
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    @pootieman I plotted the region which you described above [here](http://m.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%3E%3D4+%2Cx%5E2%2By%5E2%3C%3D16%2C+x%5E2-y%5E2%3E%3D1%2C+x%5E2-y%5E2%3C%3D9%2C+y%3E%3D-1%2C+y%3C%3D2&x=0&y=0). Perhaps this will help to simplify the integral.2012-11-04
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    The inside hyperbola starts out properly inside the inner circle, so for negative $y$ and some positive $y$ the inner hyperbola is not the left bound of the region, I think..2012-11-04

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