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

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}$$ \begin{array}{l} \Rightarrow \quad x=x^{4} \Rightarrow x\left(1-x^{3}\right)=0 \\ \Rightarrow \quad x=0 \text { and } 1=x^{3} \\ \Rightarrow \quad x=0 \text { and } x=1 \\ \text { When } x=0, \text { then } y=0 \\ \text { When } x=1, \text { then } y=1^{2}=1 \end{array} $ $\therefore$ The point of intersection is (0,0) and (1,1) $\therefore$ Area of shaded region $$ \begin{array}{l} =\int_{0}^{1}\left(y_{2}-y_{1}\right) d x \\ =\int_{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} \mathrm{sq} \text { units } \end{array} $$