Let C be the intersection of the sphere and the plane z=1, oriented counter-clockwise as viewed from high above the z-axis. Use Stokes' Theorem to evaluate.
Is my solution right?
Set
By Stokes' Theorem, the line integral is equal to the surface integral
Sigma': flat disk
so
so
Yes.
When , the equation for the surface becomes
defining a circle of radius . Our limits for in polar coordinates will therefore be
I believe that HallsofIvy was just giving the integral
which was not what I calculated for the surface integral in question. In fact, what you have tried to use is called Stoke's theorem in the plane, and while you have applied it correctly, it only works for planar surfaces. Here, we must use the general form of Stoke's theorem as the surface is curved. In this case, there is a useful theorem that
which we just proved by multiplying the unit normal by in calculating the surface integral.