The area common to the curves y=x^3 and y=√x is

$A= \int\limits_0^1\left(\sqrt{x}-x^3\right) dx$

$=\frac{2}{3}\left|x^{3 / 2}\right|_0^1-\left|\frac{x^4}{4}\right|_0^1$

$=\frac{2}{3}-\frac{1}{4}=\frac{8-3}{12}=\frac{5}{12}$