Question

The area of the region bounded by the curves ${\mathrm{y}}^2=\mathrm{x}$and $\mathrm{y}=\left|\mathrm{x}\right|$is:

• A. $\frac{1}{3}$$squnits$
• B. $\frac{1}{6}$ $squnits$
• C. $\frac{2}{3}$ $squnits$
• D. $1$$squnits$

Answer 1

The correct option is B

$\frac{1}{6}$ $squnits$

Explanation for correct option

Calculating the area of the region bounded by two curves

Given, two curves are: ${\mathrm{y}}^2=\mathrm{x},\mathrm{y}=\left|\mathrm{x}\right|$

Visualizing the two curves:

Now,

$$\begin{gathered} {\mathrm{y}}^2=\mathrm{x},\mathrm{y}=\mathrm{x} \\ \Rightarrow {\mathrm{x}}^2-\mathrm{x}=0 \\ \Rightarrow \mathrm{x}(\mathrm{x}-1)=0 \\ \Rightarrow \mathrm{x}=0,1 \\ \therefore \mathrm{P}=1 \end{gathered}$$

Therefore, the area bounded by the two curves can be written as,

$$\begin{gathered} ={\int }_0^1\sqrt{x}-xdx \\ ={\left[\frac{2x^{{\displaystyle \frac{2}{3}}}}{3}-\frac{x^2}{2}\right]}_0^1 \\ =\frac{2}{3}-\frac{1}{2} \\ =\frac{1}{6} \end{gathered}$$

Hence, the correct answer is Option (B).