question_answer

The area bounded by \[y={{\sin }^{-1}}x,x=\frac{1}{\sqrt{2}}\] and \[X-\]axis is

A) \[\frac{1}{\sqrt{2}}+1\]                

B) \[1-\frac{1}{\sqrt{2}}\]

C) \[\frac{\pi }{4\sqrt{2}}\]                               

D) \[\frac{\pi }{4\sqrt{2}}+\frac{1}{\sqrt{2}}-1\]

Correct Answer: D

Solution :

Required area = Area of rectangle \[OABC\]-Area of curve OABO \[=\frac{\pi }{4}\times \frac{1}{\sqrt{2}}-\int_{0}^{\pi /4}{\sin \text{y dy}}\] \[=\frac{\pi }{4\sqrt{2}}+[\cos y]_{0}^{\pi /4}\] \[=\frac{\pi }{4\sqrt{2}}+\cos \frac{\pi }{4}-\cos 0\] \[=\frac{\pi }{4\sqrt{2}}+\frac{1}{\sqrt{2}}-1\]