Question

Find the area of the region bounded by the two parabolas $y=x^2$ and $y^2=x$.

Answer 1

The given curves are $y=x^2$ and $y^2=x$ i.e. $y=\sqrt{x}$

To find the intersection point of two curves, let us have

$x^2=\sqrt{x}$

$⟹x^4=x$

$⟹x^4-x=0$

$⟹x(x^3-1)=0$

$⟹x=0$ and $x=1$

$⟹(0,0)$ and $(1,1)$ are the points of intersection of two curves.

$\therefore$ Area bounded by these two curves $={\int }_0^1(x^2-\sqrt{x})dx$

$={|\frac{x^3}{3}-\frac{(x{)}^{\frac{3}{2}}}{\frac{3}{2}}|}_0^1$

$=\frac{1}{3}-\frac{2}{3}=-\frac{1}{3}$

Hence, area bounded by two parabolas $y=x^2$ and $y^2=x$ is $-\frac{1}{3}$.