Question

Area of the region enclosed by $y=x^2$ and $y=\sqrt{x}$ is

• A. $1$ sq. unit
• B. $\frac{1}{2}$ sq. units
• C. $\frac{1}{3}$ sq. units
• D. cannot be determined

Answer 1

Answer: (C)

We have, $y=x^2...\left(i\right)$ a parabola with vertex $\left(0,0\right)$
and $y=\sqrt{x}$
$\Rightarrow y^2=x...\left(ii\right)$ a parabola with vertex $\left(0,0\right)$
Point of intersection of $\left(i\right)$ and $\left(ii\right)$ is $\left(0,0\right)$ and $\left(1,1\right)$

Required area = area of shaded region
$=\underset{0}{\overset{1}{\int }}\left(\sqrt{x}-x^2\right)dx={\left[\frac{2}{3}{x}^{3/2}-\frac{1}{3}{x}^3\right]}_0^1$
$=\frac{1}{3}$ sq. units