Stokes' Theorem Question

What's the value of ∫∫_S (∇x F).dS for the vector filed F= where S is the paraboloid z=9-x^2-y^2 above z=5?

By equating the equation of S and the equation of the plane, we see that S and z = 5 intersect at the circle:
9 - x^2 - y^2 = 5 ==> x^2 + y^2 = 4.

If we call this circle C, then Stokes' Theorem tells us that:
∫∫S (∇x F) dS = ∫C F • dr.

We can parametrize C positively using polar coordinates as follows:
r(θ) = <2cosθ, 2sinθ, 5>, for 0 ≤ θ ≤ 2π.
(Remember: the circle lies on the plane z = 5!)

With r'(θ) = <-2sinθ, 2cosθ, 0> and:
F[r(θ)] = F(2cosθ, 2sinθ, 5) = <10sinθ, 10cosθ, 4sinθcosθ>,

we have:
∫C F • dr = ∫C F[r(θ)] • r'(θ) dθ
= ∫ (<10sinθ, 10cosθ, 4sinθcosθ> • <-2sinθ, 2cosθ, 0>) dθ (from θ=0 to 2π)
= ∫ (-20sin^2θ + 20cos^2θ) dθ (from θ=0 to 2π)
= 20 ∫ cos(2θ) dθ (from θ=0 to 2π), since cos(2θ) = cos^2θ - sin^2θ
= 10sin(2θ) (evaluated from θ=0 to 2π)
= 0.

Therefore, ∫∫S (∇x F) dS = 0.

I hope this helps!