The answer is 9π. is it true? help me check the answer. thank you

Question

If F (x,y,z)=tan z i +(x+z)j+xsiny k and σ is the portion of the cylinder z=9-x²-y² above the xy-plane oriented so that the unit normal vector n points to the outside of the cylinder, use Stoke's theorem to evaluate ∫∫_σ(∇ x F) • n dS.

Answer 1

The boundary curve C of σ is the circle x² + y² = 9 in the xy-plane z = 0. By Stokes's theorem, your integral equals the line integral ∫_C F • dr. Since z = 0 on C, F • dr = x dy on C. So

∫_C F • dr = ∫_C x dy = Area(C) = π(3²) = 9π.