Using Green's Theorem to compute area in the complex plane

Question

I'm attempting to use Green's Theorem to express the area of a region in the complex plane in terms of a contour integral, but I'm a little confused as to how this works. I have a simple closed curve $\gamma$ with interior $D$, and I believe I'm supposed to get $$\mathrm{Area}(D)=\frac{1}{2i} \oint_\gamma \overline{z} \,dz.$$ Can anyone help me justify this?

Answer 1

$$\int_\gamma \bar{z}dz = \int_\gamma (x - iy)(dx + idy) = \int_\gamma (xdx + ydy) + i \int_\gamma( xdy - ydx)$$now hit this with stokes:$$ = \int_D d(xdx + ydy) + i\int_D d(xdy - ydx) = i \int_D 2 dx \wedge dy$$