question_answer

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

A) \[\frac{3\pi }{8}\] sq. unit

B) \[\frac{5\pi }{8}\] sq. unit

C) \[\frac{\pi }{2}\] sq. unit

D) \[\frac{\pi }{8}\] sq. unit

Correct Answer: C

Solution :

[c] required area = area of the shaded region = 4 (area of the shaded region in first quadrant) \[=4\int_{0}^{1/\sqrt{2}}{({{y}_{1}}-{{y}_{2}})dx=4\int_{0}^{1/\sqrt{2}}{(\sqrt{1-{{x}^{2}}}-x)dx}}\] \[=4\left[ \frac{1}{2}\times \sqrt{1-{{x}^{2}}}+\frac{1}{2}{{\sin }^{-1}}x-\frac{{{x}^{2}}}{2} \right]_{0}^{1/\sqrt{2}}=\frac{\pi }{2}\] sq. unit