Given curves are $y=x^{2}$ and $x=y^{2}$, which is the form of parabola.
The point of intersection, $x=\left(x^{2}\right)^{2}$
$\Rightarrow x=x^{4} $
$\Rightarrow x\left(1-x^{3}\right)=0$
$\Rightarrow x=0$ and $1=x^{3}$
$\Rightarrow x=0$ and $x=1$
When $x=0$, then $y=0$
When $x=1$, then $y=1^{2}=1$
$\therefore $ The point of intersection is $(0,0)$ and $(1,1)$.
$\therefore $ Area of shaded region
$=\int\limits_{0}^{1}\left(y_{2}-y_{1}\right) d x$
$=\int\limits_{0}^{1}\left[\sqrt{x}-x^{2}\right] d x=\left[\frac{x^{3 / 2}}{3 / 2}-\frac{x^{3}}{3}\right]_{0}^{1}$
$=\frac{2}{3}(1)^{3 / 2}-\frac{(1)^{3}}{3}-0-0$
$=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}$ sq units
