The area (in sq. units) of the region bounded by $y = 2\sqrt{1 − x^2},x ∈ [0,1]$ and x-axis is equal to

$\frac{\pi}{2}$

The correct answer is Option (3) → $\frac{\pi}{2}$

$y=2\sqrt{1-x^2},\; x\in[0,1]$

The required area is

$\int_{0}^{1}2\sqrt{1-x^2}\,dx$

$=2\int_{0}^{1}\sqrt{1-x^2}\,dx$

$\int_{0}^{1}\sqrt{1-x^2}\,dx$ represents the area of a quarter circle of radius $1$

$=\frac{\pi(1)^2}{4}=\frac{\pi}{4}$

Hence area

$=2\times\frac{\pi}{4}=\frac{\pi}{2}$

The required area is $\frac{\pi}{2}$ square units.