Use Stokes' Theorem to evaluate the line integral?

Question

Use Stokes' Theorem to evaluate the line integral where F=<(x^2)(e^x)-y, sqrt(y^2+1), z^3> and where C is the boundary of the portion of z=4-x^2-y^2 above the xy-plane.

The curl is <0, 0, 1> but what to do after that? Show your work through steps.

Answer 1

First let's find dS:

dS = (-∂z/∂x, -∂z/∂y, 1) dx dy = (2x, 2y, 1) dx dy

We also need the curl(F), which you obtained correctly as (0,0,1).

So if D is the disk {(x,y) | x²+y² ≤ 4} we get

∫_C F·ds = ∫∫_D curl(F)·dS = ∫∫_D dx dy = 4π.