There is a step in the proof of Green's Theorem where we must combine the line integrals of each curve in the same direction.
$$\oint\limits_C P(x,y)\,dx = \int_a^b P(x_1,y_1)\,dx + \int_b^a P(x_2,y_2)\,dx$$ $$\oint\limits_C Q(x,y)\,dy = \int_c^d Q(x_1,y_1)\,dy + \int_d^c Q(x_2,y_2)\,dy$$
But in order to do so, we must make the limits of integration match.
If we set the limits from $a$ to $b$ and from $d$ to $c$, the proof works as normal and we get:
$$\oint\limits_C P\,dx + Q\,dy = \iint\limits_R\left(\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}\right)\,dA$$
However, we could also set our limits from $b$ to $a$ and $c$ to $d$, but then we end up with:
\begin{align} \oint\limits_C P(x,y)\,dx &= \int_b^a P(x_2,y_2)\,dx - \int_b^a P(x_1,y_1)\,dx \\ \oint\limits_C Q(x,y)\,dy &= \int_c^d Q(x_1,y_1)\,dy - \int_c^d Q(x_2,y_2)\,dy \end{align}
Which then becomes:
\begin{align} \oint\limits_C P(x,y)\,dx &= \int_b^a P(x_2,y_2) - P(x_1,y_1)\,dx \\ \oint\limits_C Q(x,y)\,dy &= - \int_c^d Q(x_2,y_2) - Q(x_1,y_1)\,dy \end{align}
From that, you can see we will end up with:
$$\oint\limits_C P\,dx + Q\,dy = \iint\limits_R\left(\dfrac{\partial P}{\partial x} - \dfrac{\partial Q}{\partial y}\right)\,dA$$
My question is: Why does one set of limits produce the correct result while the other does not, and why would we choose one set of limits over the other (besides that it yields the result we are after)?
