How is the Fundamental Theorem of Calculus Applied to Multivariable Functions?

[CSteiner]

So for a function of a single variable

How can this be extended to the integration of the total differential of a multivariable function over a region (specifically one of two variables)?
That is, how do you integrate

Say over the circular region

,

 

[hunt_mat]

The generalisation to this to an arbitrary number of dimensions is Stokes' theorem.

 

[CSteiner]

Ah, thanks. I guess I'm going to have to get off my ass and finish that MIT OCW Multivariable Calculus course I've been studying. I'm about halfway through, so I've seen double integrals and differentials, but not stoke's theorem.