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,

and

and

.

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...