What is the best approach for solving a tricky triple integral problem?

[halcyone]

Homework Statement

A triple integral, with the bounds, from outer to inner:
integrate from -1 to 1 with respect to x
integrate from 0 to 1-x^2 with respect to y
integrate from 0 sqrt (y) with respect to z
on the function x^2*y^2*z^2

Homework Equations

none

The Attempt at a Solution

I know what kind of a region it is. The region intersects at x^2+z^2=1. However, my attempts at solving this integral lead to a messy, impossible looking integral, and I am fairly sure that this integral requires no more than integration by parts. I've tried changing the bounds, such as letting D=half-circle in xz plane, and let the bounds on y be from z^2 to 0, but they lead to similar problems. What else can I do?...

 

[Feldoh]

Try a transformation from rectangular to something else...