Area Under Folium of Descartes

The folium of |Descartes is the curve determined by the equation  
\[x^3 +y^3 =3axy\]
, with  
\[a>0\]
.

Substitute  
\[y=px\]
  to obtain  
\[x^3 + p^3 ax^23 = 3 p x^2 \rightarrow x= \frac{3ap}{1+p^3}\]

The loop starts at  
\[x=y=p=0\]
  and when p=1,  
\[x=y=\frac{3a}{2}\]

As  
\[p \rightarrow \infty\]
  the line approaches the origin along the positive  
\[y\]
  axis.
Green's equation gives
\[\begin{equation} \begin{aligned} A &= \frac{1}{2} \oint_C x \: dy - y \: dx \\ &= \frac{1}{2} \oint_C x \: (p \: dx + x \: dp - y \: dx \\ &= \frac{1}{2} \oint_C x^2 \: dp \\ &=\frac{1}{2} \int^{\infty}_0 \frac{9a^2 p^2}{(1+p^3)^2} \: dp \\ &=9 a^2 \frac{1}{2} \int^{\infty}_0 \frac{ p^2}{(1+p^3)^2} \: dp \\ &= \frac{9 a^2}{2} \int^{\infty}_0 \frac{d}{dp} (- \frac{ 1}{3(1+p^3)}) dp \\ &= 9a^2 [- \frac{1}{3 (1+p^3) }]^{\infty}_0 \\ &= \frac{3a^2}{2} \end{aligned} \end{equation}\]
 

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