The area of the region bounded by the circle x2 + y2 = 1 is:

Question

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

Smart Achievers · Accepted Answer

(2)

Ans. $π$ sq. units

Explanation:

We have, $x^{2} + y^{2} = 1^{2}$ $[∵ r = \pm 1]$

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

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

Therefore, area enclosed by the circle,

$= 2 \int_{- 1}^{1} \sqrt{1^{2} - x^{2}} d x$ $[∵ \int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x]$

$= 2 \cdot 2 \int_{0}^{1} \sqrt{1^{2} - x^{2}} d x$

$= 2 \cdot 2 {[\frac{x}{2} \sqrt{1^{2} - x^{2}} + \frac{1^{2}}{2} \sin^{- 1} (\frac{x}{1})]}_{0}^{1}$

$= 4 [\frac{1}{2} \cdot 0 + \frac{1}{2} \cdot \frac{π}{2} - 0 - \frac{1}{2} \cdot 0]$

$= 4 \cdot \frac{π}{4} = π sq. units.$

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