In Gamelin's book on Complex Dynamics, theorem 1.1 states that for $0 < r < 1$, we set $D_r = C - g(\Delta_r)$ where I used a - sign as opposed to backslash as a set difference, and the $g(z) = 1/z + b_0 + b_1z + ...$ and $\Delta_r$ is the open disk of radius $r$ centered at the origin. Then by Green's theorem,
$\text{area}(D_r) = \displaystyle\int \int_{D_r} dxdy = \frac{1}{2i}\displaystyle\int_{\partial D_r} \bar{z}dz = \frac{-1}{2i}\displaystyle\int_{\partial \Delta_r} \bar{g}dg$
Now, I understand the bounds on integration since Green's theorem relates an integral within the area to the integral on the boundary, but I don't understand how we go from the first to the second or third integral using Green's theorem. If this could be elucidated, that would be a huge help.