2) use spherical coordinates to find the integral of f(x, y,z) = z over x^2 + y^2 + z^2 < 4 and x, y,z < 0

1. Denote by the region bounded by the cylinder and the planes and . Converting to cylindrical coordinates involves setting



and is obtained by varying the parameters over , , and . The volume element is



so the integral becomes



2. Now let denote the region bounded by the sphere and the coordinate planes in the octant in which each coordinate of the point is negative.

Converting to spherical coordinates, we set



where we obtain by varying over , , and . The volume element is now



so the integral becomes