Calculate area bounded by curve using Green's theorem

Question

For homework, I need to find the area bounded by the curve $x^3+y^3=3xy$. We were given a series of hints, of which I understand very little.

First, we're told to set $y=tx$ to obtain a parametrization $y(t)=\frac{3t^2}{1+t^3},x(t)=\frac{3t}{1+t^3}$. Then, to calculate the area we're told to use the form $\omega =\frac 12(xdy-ydx)=\frac 12 x^2(y/x)^\prime dt$, since in the current case $y/x=t$ whence the form is $\frac 12x^2dt$.

I don't understand why $x(t),y(t)$ as given are a parametrization of the curve? It looks implicit and I don't see how to parametrize it.

Second of all, suppose I just go to calculate. How to calculate $\frac 12\int _0^\infty \frac{9t^2}{(1+t^3)^2}$dt?

Answer 1

To check the parametrisation is just algebra: take $y=tx$ as suggested, substitute into $x^3+y^3=3xy$, solve for $x$ in terms of $t$.

For the integral, try substituting $u=1+t^3$.