# Help with Triple Integration in Cylindrical Coordinates

• Nov 12th 2009, 02:16 PM
messianic
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
• Nov 12th 2009, 04:57 PM
Scott H
To find the boundaries of integration, we find where the two surfaces intersect. For instance,

\displaystyle \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 $\displaystyle 2$. Our integral is therefore

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