By the way, I also posted this in the Urgent Help section.
Okay, I get the half that is normally explained in the books which is the same as on wikipedia (http://en.wikipedia.org/wiki/Green's_theorem).
This is the best that I can explain this:
I understand the part of the explaination where integral of (P dx) = - double integral of (partial derivative of P with respect to y) dA.
I don't, however, understand why on the other half, integral of (Q dy) = double integral of (partial derivative of Q with respect to x) dA, does not have a negative sign in front of one of the two halves.
If essetially you are doing the same process twice, once with a type I region and the other time with a type II region, why does one half of the proof end up with a negative (which I understand) but not the other half?