(878,#9) Using cylindrical coordinates, evaluate

where E is the solid that lies within the cylinder x² + y² = 1, the plane z = 0, and below the cone z² = 4 x² + 4 y².

Solution: The domain of integration is bounded by the yellow cone, the red cylinder, and the blue disk:

Thus:

The integral is equal to (2/5) Pi:

lprint( int((r*cos(t))^2 *r, z));
lprint( int((r*cos(t))^2 *r, z = 0..2*r));
r^3*cos(t)^2*z
2*r^4*cos(t)^2

lprint( int(int((r*cos(t))^2 *r, z = 0..2*r),r));
lprint( int(int((r*cos(t))^2 *r, z = 0..2*r),r = 0 ..1));
2/5*r^5*cos(t)^2
2/5*cos(t)^2

lprint( int(int(int((r*cos(t))^2 *r, z = 0..2*r),r = 0 ..1), t ) );
lprint( int(int(int((r*cos(t))^2 *r, z = 0..2*r),r = 0 ..1), t = 0..2*Pi) );
1/5*cos(t)*sin(t)+1/5*t
2/5*Pi

by:sh