Question

The area of the region bounded by the curves $y=x^2$ and $x=y^2$ is

• A. 1/3
• B. 1/2
• C. 1/4
• D. 3

Answer 1

Answer: (A)

Given curves are $y=x^2$ and $x=y^2$, which is the form of parabola.

The point of intersection, $x={\left(x^2\right)}^2$
$\Rightarrow x=x^4$
$\Rightarrow x\left(1-x^3\right)=0$
$\Rightarrow x=0$ and $1=x^3$
$\Rightarrow x=0$ and $x=1$
When $x=0$, then $y=0$
When $x=1$, then $y=1^2=1$
$\therefore$ The point of intersection is $(0,0)$ and $(1,1)$.
$\therefore$ Area of shaded region
$=\underset{0}{\overset{1}{\int }}\left(y_2-y_1\right)dx$
$=\underset{0}{\overset{1}{\int }}\left[\sqrt{x}-x^2\right]dx={\left[\frac{x^{3/2}}{3/2}-\frac{x^3}{3}\right]}_0^1$
$=\frac{2}{3}(1{)}^{3/2}-\frac{(1{)}^3}{3}-0-0$
$=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}$ sq units