Evaluate z2 dS where S is the part of the cone z that lies

Evaluate z2 dS, where S is the part of the cone z = that lies inside the cylinder x2 + y2 = 9.

Solution

Note that these surfaces intersect when z = v(3) = v3. Parameterize S by r(u, v) = for u in [0, 1] and v in [0, p]. Since r_u x r_v = <-uv3 cos v, -uv3 sin v, u>, ||r_u x r_v|| = 2u. So, ??s z² dS = ?(u in [0, 1]) ?(v in [0, p]) (uv3)² (2u dv du) = ?(u in [0, 1]) 6pu^3 du = 6p/4