Boundary formula for planar area ($|R|=\frac{1}{2}{\oint }_{\partial R}(xdy-ydx)$)

Apply Green theorem with $P=-y/2$ and $Q=x/2$ to a positively oriented regular planar region. Since $Q_x-P_y=1$, its area is the displayed line integral. Clockwise orientation changes the sign; holes have the opposite boundary orientation to the outer curve.