calculus and analysis – Right code for solving Stoke’s Theorem problem

In solving Stoke’s Theorem problems, I have developed a step by step method. I’m now trying to write a MMA script to help me solve these problems. But I’m a little stuck.

Here’s my step by step method, which works well if tediously by hand.

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So imagine this problem:

Using Stoke’s Theorem, solve the surface integral $F(x,y,z) = Cos(z)i + x^2j + 2yk$ where $C$ is the intersection of the plane $z = 2-x$ and $x^2 + y^2 = 4$. The answer is 8$pi$.

I started my step by step with this code, but bogged down in how to make each vector element from the curl into the $P, Q, R$ of Step 2 and how then to multiply the partial derivatives by $P,Q,R$ in Step 5.

F = {Cos(z),  x^2, 2 y}
C = Curl(F, {x, y, z}) (*Step 1*)

g = 2 - x (*Step 3 solving for z of the curve*)
g1 = D(g, x) )(*Step 4*)
g2 = D(g, y)

Thanks for any help.