Question

The area enclosed between the curve $y^2=x$ and $x^2=y$ is equal to

Answer 1

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

Here, we can swet $y^2=x$ or $y=\sqrt{x}=f\left(x\right)$ and $y=x^2=g\left(x\right)$

where $f\left(x\right)\ge g\left(x\right)$ in $[0,1]$

Therefore, the required area of the shaded region

$={\int }_0^1[f\left(x\right)-g\left(x\right)]dx={\int }_0^1[\sqrt{x}-x^2]dx={[\frac{2}{3}{x}^{\frac{3}{2}}-\frac{x^3}{3}]}_0^1=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}$