(878,#5) Using cylindrical coordinates, evaluate

where E is the region bounded by the cylinder x² + y² = 4 and the planes z = -1 and z = -2.

Solution: The domain of integration is an ordinary cylinder:

Thus:

The integral is equal to 24 Pi:

lprint( int(r^2*r, r));
lprint( int(r^2*r, r = 0..2));
1/4*r^4
4

lprint( int(int(r^2*r, r = 0..2),t));
lprint( int(int(r^2*r, r = 0..2),t = 0..2*Pi));
4*t
8*Pi

lprint( int(int(int(r^2*r, r = 0..2),t = 0..2*Pi), z));
lprint( int(int(int(r^2*r, r = 0..2),t = 0..2*Pi), z = -1..2));
8*Pi*z
24*Pi

by:sh