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
Answer & explanation
Correct answer: option 3
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.