1. The problem statement, all variables and given/known data http://puu.sh/d5FpW.png [Broken] 2. Relevant equations ∫∫(∇XF)⋅dS = ∫F(r(t))⋅r'(t)dt 3. The attempt at a solution So I figured I'd have two line integrals (using Stokes's theorem. Paramatizing S1: I can think of the cylinder as the closed curve circle x²+y² = 9 on the x-y plane right? x = 3cost, y = 3sint, z = 0; 0 ≤ t ≤ 2pi or r(t) = (3cost)i + (3sint)j Then F = (27(cost)^3)i +(2187(sint)^7)j The dot product of F and r'(t) gives me -81sint(cost)3 + 6561(sint)^7cost Then I integrate this from t = 0 to t = 2pi and get 0. For S2: Similar to S1, I can think of this hemisphere as the circle x²+y² = 9 on the x-y plane with z = 8 right? Then paramatizing: r(t) = (3cost)i + (3sint)j + 8k ; 0≤t≤2pi I go through the same process as S1, but I don't get 0 I get -576 pi, which is obviously wrong. Would someone mind walking me through how to do this problem correctly?Am I applying Stokes's theorem incorrectly.