The area of the region bounded by the circle \(x^2+y^2=1\) is _________ sq. units.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\pi\)
The circle has centre (0, 0) and radius 1. It is symmetric about both axes, so its area is 4 times the area of the quarter in the first quadrant, under \(y = \sqrt{1 - x^2}\) from x = 0 to x = 1:
\( \displaystyle \text{Area} = 4\int_{0}^{1}\sqrt{1 - x^2}\,dx \)
\( \displaystyle = 4\left[\frac{x}{2}\sqrt{1 - x^2} + \frac{1}{2}\sin^{-1}x\right]_{0}^{1} \)
\( \displaystyle = 4\left(0 + \frac{1}{2}\cdot\frac{\pi}{2} - 0\right) \)
\(= \pi\)
This agrees with the formula \(\pi r^2\) for r = 1.
NCERT Exemplar:
Application of Integrals
