Let $G$ be the solid in the first octant enclosed by the surfaces $z = x^2 + y^2$ and $z = 36 - 3x^2 - 3y^2.$ Use cylindrical coordinates to evaluate: $$\iiint x^2+y^2\,dV.$$
I'm not entirely sure how to set up the limits of integration. I know the integration order will be $dz\,dr\,d\theta$. So far what I have is:
$$\int_{r^2}^{3(12-r^2)} \, dz$$
Any help would be much appreciated!
Where do the two surfaces intersect?
$$x^2 + y^2 = 36 - x^2 - y^2\\ x^2 + y^2 = 9$$
$$r = 3$$
$$\int_0^{2\pi}\int_0^3 (36 - 4r^2) r\ dr\ d\theta$$