Calculate the integral s f · ds, where s is the entire surface of the solid half ball x2 + y2 + z2 ≤ 1, z ≥ 0, and f = (x + 3y5)i + ( y + 10xz)j + (z − xy)k. (let s be oriented by the outward-pointing normal.)

The divergence theorem applies since is a closed surface. We have



where is the space enclosed by the surface . The divergence of the given vector field is



So we can write the volume integral (in spherical coordinates) as