 Solution

Let $D$ be a bounded planar region with a piecewise smooth, simple closed boundary $C$, oriented counterclockwise, and let $P,Q$ have continuous first partial derivatives on an open neighbourhood of its closure. Green's theorem states thatFor a region with holes, orient every boundary component with $D$ on its left: the outer component is counterclockwise and the inner components are clockwise.

Choose $P=-y/2$ and $Q=x/2$. Their indicated partial derivatives differ by one, so the area isThis is signed area as a boundary integral; reversing the direction of $C$ changes the sign of the integral.