Question

Choose the correct answer in the following.

Area lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and the lines x=0 and x=2

(a) $\pi$ (b) $\frac{\pi }{2}$

(c) $\frac{\pi }{3}$ (d) $\frac{\pi }{4}$

Answer 1

The area bounded by the circle and the lines x = 0 and x = 2, in the first quadrant is represented in the figure by shaded region.

Required area $={\int }_0^2|y|dx={\int }_0^2\sqrt{4-x^2}dx$

$=[\frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}si{n}^{-1}(\frac{x}{2}){]}_0^2=0+2si{n}^{-1}(1)-0=2\times \frac{\pi }{2}=\pi squnit$
Thus, the correct option is (a).