Hey guys. I'm stuck on a problem for my calc class.

Let D be the region bounded by the paraboloids $\displaystyle z = 2 - x^2 - y^2$ and $\displaystyle z = 2x^2 + y^2$. Find the volume of the region D using an iterated integral.

I have been trying all night to come up with the limits of integration and the general set up, and I am at a loss.

So far I have tried to solve by subbing z for itself:

$\displaystyle 2 - x^2 - y^2 = 2x^2 + y^2$

but this does not leave a clean answer to transform into spherical.

you're left with:

$\displaystyle 3x^2 + 2y^2 = 2$, and I can't see any way to get this into a nice $\displaystyle x^2 + y^2 = r^2$ format.

We have a test coming up and my professor said a problem like this would be on it. Any help would be greatly appreciated.