Question

The area of the region bounded by two parabolas $\mathrm{y}={\mathrm{x}}^2$ and $\mathrm{x}={\mathrm{y}}^2$ is

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

Answer 1

The correct option is B

$\frac{1}{3}$

Explanation for the correct option:

Given,

$\mathrm{y}={\mathrm{x}}^2$ and $\mathrm{x}={\mathrm{y}}^2$

Solving the above two curves simultaneously, we get

Intersection point as $\left(1,1\right)$ and $\left(0,0\right)$

So,

Area,

$$\begin{gathered} A={\int }_0^1\left(y_{uppercurve-}{y}_{lowercurve}\right)dx \\ ={\int }_0^1\left(\sqrt{x}-x^2\right)dx \\ ={\left[\frac{2}{3}{x}^{\frac{2}{3}}-\frac{x^3}{3}\right]}_0^1 \\ =\frac{2}{3}-\frac{1}{3} \\ =\frac{1}{3} \end{gathered}$$

Hence, option $\left(\mathrm{B}\right)$ is the correct answer