Question

The area (in sq. units) of the region enclosed by the curves $y=x^2$ and $y=\sqrt{|x|}$ is

• A. $\frac{4}{3}$
• B. $\frac{1}{6}$
• C. $\frac{1}{3}$
• D. $\frac{2}{3}$

Answer 1

The correct option is D $\frac{2}{3}$

Here, curves are $y=x^2$
and $y=\sqrt{|x|}=\{\begin{array}{cc}\sqrt{x},& x\ge 0\\ \sqrt{-x},& x<0\end{array}$


$y=x^2$
$\Rightarrow \sqrt{x}=x^2$
$\Rightarrow x=x^4$
$\Rightarrow x(x^3-1)=0$
$\Rightarrow x=0,1$

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