The area of the region bounded by parabola y2 = x and the straight line 2y = x is:

Question

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

Smart Achievers · Accepted Answer

(1)

Ans. $\frac{4}{3}$ sq. units

Explanation :

We have to find the area enclosed by parabola $y^{2} = x$ and the straight line $2 y = x .$

$∴$ ${(\frac{x}{2})}^{2} = x$

$\Rightarrow$ $x^{2} = 4 x$

$\Rightarrow$ $x (x - 4) = 0$

$\Rightarrow$ $x = 4$

$\Rightarrow$ $y = 2$

and $x = 0$

$\Rightarrow$ $y = 0$

So, the intersection points are (0, 0) and (4, 2).

Area enclosed by shaded region,

$A = \int_{0}^{4} (\sqrt{x} - \frac{x}{2}) d x$

$= {[\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} - \frac{1}{2} \cdot \frac{x^{2}}{2}]}_{0}^{4}$

$= {[2 \cdot \frac{x^{3 / 2}}{3} - \frac{x^{2}}{4}]}_{0}^{4}$

$= \frac{2}{3} {(4)}^{3 / 2} - \frac{16}{4} - \frac{2}{3} (0) + \frac{1}{4} (0)$

$= \frac{16}{3} - \frac{16}{4} = \frac{64 - 48}{12} = \frac{16}{12} = \frac{4}{3} sq. units.$

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