# Volume of Bounded region

• Nov 22nd 2010, 12:07 AM
moderata
Volume of Bounded region
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.
• Nov 22nd 2010, 12:37 AM
moderata
even just using (per Allan Cuz's notes): \$\displaystyle 2 - x^2 - y^2 = 2 - r^2\$ and \$\displaystyle 2x^2 + y^2 = 2r^2\$ you still end up with a weird result: \$\displaystyle 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.