= 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 $Q=x$ gives
$$
\oint_C(x\,dy-y\,dx)=\iint_D\left(\frac{\partial x}{\partial x}-\frac{\partial(-y)}{\partial y}\right)dx\,dy=2\iint_Ddx\,dy=2\mathcal A.
$$
The 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 $x=a\cos\theta$, $y=b\sin\theta$, $0\le\theta\le2\pi$. Then
$$
x\,dy-y\,dx=ab\cos^2\theta\,d\theta+ab\sin^2\theta\,d\theta=ab\,d\theta.
$$
Hence $2\mathcal A=\int_0^{2\pi}ab\,d\theta=2\pi ab$, and
$$
\boxed{\mathcal A=\pi ab}.
$$
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