Find the area of the region bounded by the two parabolas $y=x^2 $ and $y^2=x$ :

$\frac{1}{2}$ $\frac{1}{3}$ $\frac{2}{3}$ $\frac{5}{3}$

$\frac{1}{3}$

The correct answer is Option (2) → $\frac{1}{3}$ $y=x^2 $ and $y^2=x$ they both intersect at $x^4=x$ $⇒x=0,1$ Required area = $\int\limits_0^1\sqrt{x}-x^2dx$ $=\left[\frac{2}{3}x^{5/2}-\frac{x^3}{3}\right]_0^1=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}$