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Q.
Area of the region enclosed by $y=x^{2}$ and $y=\sqrt{x}$ is
Application of Integrals
Solution:
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
$=\int\limits_{0}^{1}\left(\sqrt{x}-x^{2}\right)dx =\left[\frac{2}{3}x^{3 /2}-\frac{1}{3}x^{3}\right]_{0}^{1}$
$=\frac{1}{3}$ sq. units
