Find [tex]\int\int\int y^2 z^2[/tex]where E is the region bounded by the paraboloid x = 1 - y22 - z2 and the plane x = 0.
The Attempt at a Solution
The region is a paraboloid with vertex at x = 1, y = 0, z = 0. I chose z bounds to be between 0 and 1 - y22 - z2 for first integral. Then, I realized that since region was bounded by x = 0 plane, the y and z values would range (in polar coordinates), from 0 to 2[tex]\pi[/tex] for y (or z) and 0 to 1 for z.
Then, upon finishing first integral for z bounds, I got (1 - y2 - z2) * y2*z2, and when converting to polar coordinates, I got,
(1 - r2)*(r4*cos2([tex]\theta[/tex])*sin2([tex]\theta[/tex])
I don't know how to simplify this expression so that I can integrate for theta. How do I do it?