 Solution

Take $C$ to be a simple closed, piecewise smooth curve oriented counterclockwise around its interior $D$. Green's theorem applied to $P=-y$ and $Q=x$ givesThe orientation is essential: a clockwise unit circle gives $-2\pi$, whereas twice its ordinary unsigned area is $+2\pi$. For a nonsimple curve, the corresponding formula concerns signed area with winding multiplicity rather than one unsigned enclosed area.

For positive semiaxes $a,b$, parametrize the ellipse counterclockwise by $x=a\cos \theta$, $y=b\sin \theta$, $0\le \theta \le 2\pi$. ThenHence $2\mathcal{A}={\int }_0^{2\pi }abd\theta =2\pi ab$, and