(949, #9) Evaluate

where S is the hemisphere x² + y² + z² = 4, z >= 0.

Solution: Use the spherical-coordinate parametrization

x = 2 sin(u) cos(v), y = 2 sin(u) sin(v) z = 2 cos(u), u = 0..Pi/2 , v = 0 ..2Pi.

Then (see text, p935)

Thus the integral above is equal to