Evaluate the surface integral ∫∫S F · dS.

Question

Evaluate the surface integral ∫∫_S F · dS.

F(x, y, z) = y j − z k, S is the closed surface that consists of the paraboloid y = x² + z², 0 ≤ y ≤ 1 and the disk x² + z² ≤ 1, y = 1.

Answer 1

On the paraboloid, dS = ndxdy where the outward unit normal n is in the direction of ∇(x² – y + z²) = 2xi – j – 2zk. On the disk, dS = ndxdz where n = j. That should get you most of the way to the solution.