The area of the region bounded by the curves $y = x^2$ and $y=x$ is

$y = x^2$ represent a parabola opening upward and $y = x$ is a straight line passing through origin. The two curves are meeting at $(0, 0)$ and $(1, 1)$.Required area $={\int }_0^1 \left(x-x^2\right)dx$
                         $=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$