question_answer

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

A)  \[\left( \frac{1}{\sqrt{2}}+1 \right)sq\,unit\]

B)  \[\left( 1-\frac{1}{\sqrt{2}} \right)sq\,unit\]

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

D)  \[\left( \frac{\pi }{4\sqrt{2}}+\frac{1}{\sqrt{2}}-1 \right)sq\text{ }unit\]

E)  None of the above

Correct Answer: D

Solution :

Required area = Area of rectangle OABC - Area of curve OBCO \[=\frac{\pi }{4\sqrt{2}}-\int_{0}^{\pi /4}{\sin y\,dy}\] \[=\frac{\pi }{4\sqrt{2}}+[\cos y]_{0}^{\pi /4}\] \[=\left[ \frac{\pi }{4\sqrt{2}}+\left( \frac{1}{\sqrt{2}}-1 \right) \right]sq\,unit\]