Green's Theorem: Explaining the Bounds Reversal

[TysonM8]

I’m having a little trouble understanding why Green’s Theorem is defined as;

∮_C P dx+Q dy = ∬_D [(δQ/δx)-(δP/δy)] dA

Instead of;

∮_C P dx+Q dy = ∬_D [(δQ/δx)+(δP/δy)] dA

When proving the theorem, in the first step you simply reverse the bounds of the second integral to get the result;

∮_C P dx = -∫_(x=a)^(x=b) ∫_(y=g_1 (x))^(y=g_2 (x)) δP/δy dydx

But in the second step, the bounds are kept how they are to keep the double integral positive. So you have;

∮_C Q dy = ∫_(y=a)^(y=b) ∫_(x=h_2 (y))^(x=h_1 (y)) δP/δy dxdy

So can anyone explain why the bounds were reversed in the step?

 

[HallsofIvy]

Because, in the proof, you reverse the direction along the curve:

You divide the closed curve at two points, t_0 and t_1, and integrate along the top and bottom curves from t_0 to t_1. If you are going counterclockwise around the curve over the top half, then you are going clockwise over the bottom half. In order to have a single integration around the full curve, you have to reverse the direction of one half.

 

[TysonM8]

That's the first time you reverse the bounds (you also do this for the second step, reversing the bounds a and b), but you reverse the bounds of the second integral (g_1(x) and g_2(x)) later on. The question is, why isn't this done for the second step as well?