A surprising result from Green's theorem

Question

I use Green's theorem to compute:

$$ \int^{1,3}_{-1,2} y^2 dx+2xy dy $$

and I got 0 as $\partial_x F_2$ cancels out $\partial_y F_1$.

Then I try integrate along the line segment from $(-1,2)$ to $(1,3)$ I got 13.

Could anyone spot what could be wrong using Green's theorem to compute this above integral?

Answer 1

Green's theorem only applies to simple closed curves, that is, curves that do not cross themselves and are closed. A line segment is not a simple closed curve.