- #1

- 36

- 0

## Homework Statement

Find [tex]\int\int\int y^2 z^2[/tex]where E is the region bounded by the paraboloid x = 1 - y

^{2}2 - z

^{2}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 - y

^{2}2 - z

^{2}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 - y

^{2}- z

^{2}) * y

^{2}*z

^{2}, and when converting to polar coordinates, I got,

(1 - r

^{2})*(r

^{4}*cos

^{2}([tex]\theta[/tex])*sin

^{2}([tex]\theta[/tex])

**I don't know how to simplify this expression so that I can integrate for theta. How do I do it?**