Green's theorem provides an elegant way to understand the connection between the ideas of line integrals around closed curves and double integrals over regions. In particular, we may use Green's theorem to calculate areas.
Suppose we want to calculate
$A=\displaystyle\frac{1}{2}\oint_{\mathcal{C}} xdy-ydx$,
in which the curve $\mathcal{C}$ is an oriented ellipse of equation $x^{2}+2xy+3y^{2}=4$. As one can readily see, the vertices of the ellipse are $(\pm2,0)$ and $\displaystyle\left(0,\pm \frac{2}{\sqrt{3}}\right)$.
In order to evaluate the area $A$ through the Green theorem, how do I parametrize the equation $x^{2}+2xy+3y^{2}=4$ using polar coordinates?