Question

Area of the region bounded by two parabolas $y=x^2$ and $x=y^2$ is

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

Answer 1

Answer: (A)

The intersection point of parabolas $y=x^2$ and $x=y^2$ is
$y={\left(y^2\right)}^2$
$\Rightarrow y=y^4$
$\Rightarrow y=0,y=1$
$\Rightarrow x=0,x=1$

So, the intersection point in $O(0,0)$ and $(1,1)$.
$\therefore$ Required area $=\underset{0}{\overset{1}{\int }}\left(y_2-y_1\right)dx$
$=\underset{0}{\overset{1}{\int }}\left(\sqrt{x}-x^2\right)dx$
$={\left[\frac{x^{3/2}}{3/2}-\frac{x^3}{3}\right]}_0^1={\left[\frac{2}{3}{x}^{3/2}-\frac{x^3}{3}\right]}_0^1$
$=\left[\frac{2}{3}(1)-\frac{1}{3}(1)+0-0\right]$
$=\left[\frac{2}{3}-\frac{1}{3}\right]=\frac{1}{3}$
Alternate method
We know that, if parabolas are $y^2=4ax$ and
$x^2=4by$, then area of bounded region is $\frac{4a\cdot 4b}{3}.$
$\therefore$ Required area $=\frac{1\times 1}{3}=\frac{1}{3}$