as you found, is the correct integral.Evaluate:
where V is the region bounded by the paraboloid and the -plane.
My sol'n (but I am unsure, because it's ugly!):
[Ah, just realised I missed an 'r' when I was integrating as I wrote this. The solution is now simpler, but the computation is very long]
When z=0, the region R in the xy-plane is the circle with radius=2. So so in polar coords,
So we have: (I integrated along Z first, don't know if that's right)
Skipping boring computation (the second time I did it in maxima ), we get:
Is that right?
EDIT: Hm... looking at the integrand now, I'm thinking I should have left the z in there, and not substituted it for 4-r^2...