# Area between curves.

I have to calculate area bounded by curves : $$(x^3+y^3)^2=x^2+y^2$$ for $$x,y \ge 0$$. I tried to use polar coordinates, but I have : $$r^4(\cos^6\alpha +2\sin^3\alpha\cos^3\alpha + \sin^6\alpha)=1$$

By your work: $$r=\frac{1}{\sqrt{\sin^3\alpha+\cos^3\alpha}}.$$ Since it's symmetric in respect to $$y=x$$, we obtain that the needed area it's $$2\int\limits_0^{\frac{\pi}{4}}d\alpha\int\limits_0^{\frac{1}{\sqrt{\sin^3\alpha+\cos^3\alpha}}}rdr=\int_0^{\frac{\pi}{4}}\frac{1}{\sin^3\alpha+\cos^3\alpha}d\alpha.$$ Can you end it now?