# Math Help - Help with Triple Integration in Cylindrical Coordinates

1. ## Help with Triple Integration in Cylindrical Coordinates

I'm having trouble with the following problems:

1.) Find the volume of the region bounded by the paraboloids z = 2x^2 +y^2 and z= 12 -x^2 -2y^2

2.) Find the volume of the region bounded above by the spherical surface x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 +y^2

Can someone give an explanation of what the boundaires would be for triple integration in cylindrical coordiantes

2. To find the boundaries of integration, we find where the two surfaces intersect. For instance,

\begin{aligned}
2x^2+y^2&=12-x^2-2y^2\;\;\;\;\;\mbox{when}\\
3x^2+3y^2&=12\\
x^2+y^2&=4,
\end{aligned}

which means that our two paraboloids intersect in a circle of radius $2$. Our integral is therefore

$\int_0^{2\pi}\int_0^2\int_{2r^2\cos^2\theta+r^2\si n^2\theta}^{12-r^2\cos^2\theta-2r^2\sin^2\theta}r\,dz\,dr\,d\theta.$