question_answer

The area of the region described by \[A=\left\{ (x,y):{{x}^{2}}+{{y}^{2}}\le 1\,and\,{{y}^{2}}\le 1-x \right\}\]is:

A) \[\frac{\pi }{2}+\frac{4}{3}\]

B) \[\frac{\pi }{2}-\frac{4}{3}\]

C) \[\frac{\pi }{2}-\frac{2}{3}\]

D) \[\frac{\pi }{2}+\frac{2}{3}\]

Correct Answer: A

Solution :

\[A=\frac{1}{2}\times \pi +2\int\limits_{0}^{1}{\sqrt{1-xdx}}\]

\[=\frac{\pi }{2}+\frac{4}{3}.\]