The area (in sq. units) of the region in the first quadrant bounded by the lines x = 2 and y = 2 and the circle \(x^{2}+y^{2}=4\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(4-\pi\)
The circle has radius 2 and touches the line x = 2 at (2, 0) and the line y = 2 at (0, 2). The region is the corner of the square \(0 \le x \le 2,\ 0 \le y \le 2\) lying outside the quarter circle. For each x its top is y = 2 and its bottom is the circle \(y = \sqrt{4-x^{2}}\):
\( \displaystyle A = \int_{0}^{2}\left(2-\sqrt{4-x^{2}}\right)dx \)
\( \displaystyle = \int_{0}^{2}2\,dx-\int_{0}^{2}\sqrt{4-x^{2}}\,dx \)
The first integral is the square, 4. The second is the quarter of the disc of radius 2, \(\frac{1}{4}\pi(2)^{2} = \pi\):
\(A = 4-\pi\)