(879,#11) Find the volume of the region E bounded by the paraboloids z = x² +y² and z = 36 - 3 x² - 3 y² .

Solution: The solid of integration, below, is produced by the hundreds at Santa's Christmas Tree Ornament Factory.

The surfaces meet when x² + y² = 36 - 3(x² + y² ). That is, when x² + y² = 9.

The volume of the ornament is:

Integrating we get 162 Pi, which is a huge ornament:

lprint( int(r,z));
lprint( int(r,z = r^2 ..36 - 3*r^2));
r*z
36*r-4*r^3

lprint( int(int(r,z = r^2 ..36 - 3*r^2),r));
lprint( int(int(r,z = r^2 ..36 - 3*r^2),r = 0..3));
18*r^2-r^4
81

lprint( int(int(int(r,z = r^2 ..36 - 3*r^2),r = 0..3),t));
lprint( int(int(int(r,z = r^2 ..36 - 3*r^2),r = 0..3),t = 0..2*Pi));
81*t
162*Pi

by:sh