Area of Loop of Folium of Descartes

Theorem
Consider the folium of Descartes $F$, given in parametric form as:
 * $\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$

The area $\AA$ of the loop of $F$ is given as:


 * $\AA = \dfrac {3 a^2} 2$

Proof
From Behaviour of Parametric Equations for Folium of Descartes according to Parameter we have that the loop is traversed for $0 \le t < +\infty$.

We convert the parametric equation to polar form:

Then we have:

Then:

which leads us to:

Hence the result.