Stoke's theorem

Hi, can anyone please help me with this question?

Let S be the capped cylindrical surface given by the union of the two surfaces
S1: $x^2+y^2=1, 1 \leq z \leq 3$
S2: $z=4-x^2-y^2, 3 \leq z \leq 4$
Let $\vec{F}=-yz^3i+2xj-3y^2x^5k$. Evaluate $\int \int (\nabla \times F) \cdot dS$ using
(a) an appropriate line integral
(b) the simplest surface for S

I have been working on this question for a long time.
For part (a),I parametrized line which is the circle, then put in vector field F and then dotted with velocity vector, and integrate t from 0 to $2\pi$, but I don't know what happened, I always end up 0!
For part (b),clearly the simplest surface would be the circle, but I still could not set up integrals and find proper normal vector.

Can anyone please help me and show me the details? Thanks a lot.