The area of the region bounded by parabola ${y^2} = x$ and the straight line $2y = x$ is

• $\frac{4}{3}$ sq units

• 1 sq units

• $\frac{2}{3}$ sq units

• $\frac{1}{3}$ sq units

Correct Option (a)

We have ${y^2} = x$ and $2y = x$
Solving, we get ${y^2} = 2y$

$\Rightarrow$ $y = 0,2$
When $y = 0,x = 0$ and when $y = 2,x = 4$

So, points of intersection are (0,0) and (4,2)

Graphs of parabola ${y^2} = x$ and $2y = x$ are as shown in the following figure.

From the figure, area of the shaded region,
$A = \int_0^4 {\left[ {\sqrt x - \frac{x}{2}} \right]} dx$

$= \left[ {\frac{2}{3}{x^{3/2}} - \frac{1}{2} \cdot \frac{{{x^2}}}{2}} \right]_0^4 = \frac{2}{3} \cdot {4^{3/2}} - \frac{{16}}{4} - 0$

$= \frac{{16}}{3} - 4 = \frac{4}{3}$sq. units