(873, #15) Evaluate the integral

where E is bouded by the paraboloid x = 4 y² + 4 z² and the plane x = 4.

Solution: The boundary of the domain of integration is displayed below. For each fixed x, y and z fall on acircle of radius sqrt(x)/2.

Consequently, with C(x) = { (x,y,z) : x² + z² <= sqrt(x)/2}:

The integral is equal to 16 Pi/3:

lprint( int(x*r , r));
lprint( int(x*r , r = 0 ..sqrt(x)/2));
1/2*x*r^2
1/8*x^2

lprint( int(int(x*r , r = 0 ..sqrt(x)/2), t));
lprint( int(int(x*r , r = 0 ..sqrt(x)/2), t = 0..2*Pi));

1/8*x^2*t

1/4*Pi*x^2

lprint( int(int(int(x*r , r = 0 ..sqrt(x)/2), t = 0..2*Pi), x));

lprint( int(int(int(x*r , r = 0 ..sqrt(x)/2), t = 0..2*Pi), x = 0..4));

1/12*Pi*x^3
16/3*Pi