find the area bounded by y^(2/3)+x^(2/3)=a^(2/3) and co-ordinate axes.
Use the parametrization $\displaystyle x=a\cos^3(t) $ and $\displaystyle y=a\sin^3(t)$ to complete the simple close.
By Greene's theorem we get
$\displaystyle \frac{1}{4}\iint_DdA=-\frac{1}{4}\oint y\,dx=-a^2\int_{0}^{2\pi}\sin^3(t)(3\cos(t)\sin(t))dt=-\frac{3a^2}{4}\int_{0}^{2\pi}\sin^{4}(t)\cos(t)\,d t $