let $A$ be a region which is the interior of $C^1$ curve $C$ (oriented counterclockwise), such that $$C(t)=(g_1(t), g_2(t)).$$
and $N$ is normal to the curve i.e. $$N(t)=(g_2'(t), -g_1'(t)).$$ you can check that easily by dotting $C$ with $N$ and getting $0$.
show that if $F=(P,Q)$ is a vector field on $A$, then $$\iint_A (\operatorname{div} F)\,dy\,dx=\int_a^bF\cdot N\,dt.$$
I don't really know how to prove this, I tried to use the fact that $$\operatorname{div} F=\nabla\cdot F=\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$$ but it didn't help at all, then I tried to evaluate $F\cdot N$ which also didn't help.