I guess I can see it in cylindrical:
z is going...
from r^2 (the paraboloid) up to 6y i.e. 6r sin theta (the plane roof)
... for every bit that r is going...
from zero (the z axis) along to as far as where r^2 = 6r sin theta => r = 6 sin theta
... for every bit that theta is going from zero (the +ive x axis) round to pi (-ive). So,
int from 0 to pi int from 0 to 6sin theta int from r^2 to 6rsin theta rz dz dr dtheta - Wolfram|Alpha
The integral graph there (going 0 to pi on the horizontal) looks about right.
Those zero sins aren't going to work, but see also,
integrate 1944 sin^6 theta - Wolfram|Alpha
which gives 121.5 pi. If this isn't wrong I'll put a pic for the computation similar to
when I'm home, could be a while.
Anyway, hope that helps.
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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