Find the area of the region bounded by $y = \sqrt{x}$ and $y = x$.

$\frac{1}{6}$ square units

The correct answer is Option (3) → $\frac{1}{6}$ square units

Given equation of curves are $y = \sqrt{x}$ and $y = x$.

On solving both equations, we get

$\Rightarrow x = \sqrt{x} \Rightarrow x^2 = x$

$\Rightarrow x^2 - x = 0 \Rightarrow x(x - 1) = 0$

$\Rightarrow x = 0, 1$

$∴$ Required area of shaded region

$= \int\limits_{0}^{1} (\sqrt{x}) \, dx - \int\limits_{0}^{1} x \, dx$

$= \left[ 2 \cdot \frac{x^{3/2}}{3} \right]_{0}^{1} - \left[ \frac{x^2}{2} \right]_{0}^{1}$

$= \frac{2}{3} \cdot 1 - \frac{1}{2} = \frac{4 - 3}{6} = \frac{1}{6} \text{ sq. unit}$