Volume of Bounded region

**moderata** · November 22nd 2010, 12:07 AM

Hey guys. I'm stuck on a problem for my calc class.
Let D be the region bounded by the paraboloids $z = 2 - x^2 - y^2$ and $z = 2x^2 + y^2$ . 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:
$2 - x^2 - y^2 = 2x^2 + y^2$
but this does not leave a clean answer to transform into spherical.
you're left with:
$3x^2 + 2y^2 = 2$ , and I can't see any way to get this into a nice $x^2 + y^2 = r^2$ 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.

**moderata** · November 22nd 2010, 12:37 AM

even just using (per Allan Cuz's notes): $2 - x^2 - y^2 = 2 - r^2$ and $2x^2 + y^2 = 2r^2$ you still end up with a weird result: $3r^2 = 2$

I am just totally lost by the integral here. I am sure it will be a triple integral, but I have no idea what to do.

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