Evaluate ∫∫S F · dS, when F(x, y, z) = 4yi + 4xj + 4zk and S is the boundary of the solid region E enclosed by the paraboloid z = 36 - x2 - y2 and the plane z = 0. SOLUTION S consists of a parabolic top surface S1 and a circular bottom surface S2. Since S is a closed surface, we use the convention of positive (outward) orientation. This means that S1 is oriented upward and we can use Equation 10 with D being the projection of S1 on the xy-plane, namely, the disk x2 + y2 ≤ 1. So we have