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

$\frac{1}{3}$

The correct answer is Option (3) → $\frac{1}{3}$

The point of intersection of these two parabolas are $O(0, 0)$ and $A(1, 1)$ as shown in the Fig.

Here, we can set $y^2 = x$ or $y = \sqrt{x} = f(x)$ and $y = x^2 = g(x)$, where, $f(x) \geq g(x)$ in $[0, 1]$.

Therefore, the required area of the shaded region

$= \int\limits_{0}^{1} [f(x) - g(x)] dx$

$= \int\limits_{0}^{1} [\sqrt{x} - x^2] dx = \left[ \frac{2}{3}x^{\frac{3}{2}} - \frac{x^3}{3} \right]_{0}^{1}$

$= \frac{2}{3} - \frac{1}{3} = \frac{1}{3}$