Find the volume of a region using an iterated integral

**downthesun01** · November 22nd 2010, 10:58 PM

The region Q is bounded by $z=2x^2+y^2$ and $z=2-x^2-y^2$

Ok, I tried using Cartesian coords and got a big mess. Then I tried changing everything to polar coords and got this:

$2r^2cos^2(\theta)+r^2sin^2(\theta)=2-r^2$

I was hoping that the left side would turn out as nicely as the right side did. Did I do something wrong when converting? Is there something I'm missing? Thanks

**HallsofIvy** · November 23rd 2010, 03:50 AM

Well, you could write $2r^2cos^2(\theta)+ r^2 sin^2(\theta)= r^2 cos^2(\theta)+ r^2(cos^2(\theta)+ sin^2(\theta))$ $= r^2(cos^2(\theta)+ 1)$ and then the equation becomes
$r^2(cos^2(\theta)+ 1)= 2- r^2$

$r^2(cos^2(\theta)+ 2)= 2$

$r= \frac{\sqrt{2}}{\sqrt{cos^2(\theta)+ 2}}$

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