(879,#35) Evaluate the integral

by changing to spherical coordinates.

Solution: Working through the ranges of integration we have, first, 0 <= x <= 3. Next, -sqrt(9-x²) <= y <= sqrt(9-x²) , which implies that x² + y² <= 9. Finally, we have 0 <= z <= sqrt(9 - x² + y²), which implies that x² + y² + z² <= 9.

Consequently, the domain of integration is the hemisphere of radius 3 centered at the origin.

Evaluating,

by:sh