The area of the region bounded by the circle ${x^2} + {y^2} = 1$ is

• $2\pi$ sq units

• $\pi$ sq units

• $3\pi$ sq units

• $4\pi$ sq units

Correct Option (b)

We have, ${x^2} + {y^2} = 1,$ which is circle
having centre at (0,0) and radius '1’

$\Rightarrow$ ${y^2} = 1 - {r^2}$
$\Rightarrow$ $y = \sqrt {1 - {x^2}}$

From the figure, area of the shaded region
$= 4\int_0^1 {\sqrt {{1^2} - {x^2}} } dx$

$= 4\left[ {\frac{x}{2}\sqrt {{1^2} - {x^2}} + \frac{{{1^2}}}{2}{{\sin }^{ - 1}}\frac{x}{1}} \right]_0^1$

$= 4\left[ {0 + \frac{1}{2} \cdot \frac{\pi }{2} - 0 - 0} \right]$

$= \pi$ sq. units