The area of the region bounded by the curves y2 = x and y = x is:

$y^2=x,\;y=x.$

$x=y^2.$

$\text{Points of intersection:}$

$y=y^2.$

$y(y-1)=0.$

$y=0,1.$

$\text{Area}=\int_{0}^{1}(y-y^2)\,dy.$

$=\left[\frac{y^2}{2}-\frac{y^3}{3}\right]_{0}^{1}.$

$=\frac12-\frac13.$

$=\frac16.$

$\text{Area}=\frac16.$