# Thread: Find the volume of a region using an iterated integral

1. ## Find the volume of a region using an iterated integral

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

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

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

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

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

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

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