complex integral of conjugate on polygonal curve

Question

I need help with solving the following question:

Let $G$ be an open, bounded and connected set which its boundary is a polygonal chain. Calculate:

$$\int_\gamma \overline z \, dz$$

where $\gamma$ is the boundary of $G$.

Any ideas or hints?

EDIT: Ok so I know that $\int_\gamma \overline z \, dz = 2iArea(G)$ but I don't know how to prove it..

Answer 1

You can write $\displaystyle\int_\gamma \overline z \, dz=\displaystyle\int_\gamma (x-iy) \, (dx+idy)=\displaystyle\int_\gamma xdx +ydy+i\displaystyle\int_\gamma -ydx +xdy$ and apply the Green's theorem.