question_answer

The area bounded by curves \[y=cosec\text{ }x,\]\[y=sec\text{ }x,\] \[y=cos\text{ }x\]and \[y=sin\text{ }x\]on interval \[\left( 0,\frac{\pi }{2} \right)\] ? is equal to

A) \[1+{{\log }_{e}}\left( \sqrt{2}+1 \right)\] 

B)      \[2{{\log }_{e}}\left( \sqrt{2}+1 \right)\]   

C)       \[2{{\log }_{e}}\left( \sqrt{2}+1 \right)-\sqrt{2}\]    

D) \[{{\log }_{e}}\left( \sqrt{2}+1 \right)-\frac{1}{\sqrt{2}}\]

Correct Answer: C

Solution :

[c] Required area, \[A=2\int\limits_{0}^{\frac{\pi }{4}}{(\sec x-\cos x)\,dx}\] \[=2\left[ {{\log }_{e}}|\sec x+\tan x|-\sin x \right]_{0}^{\pi /4}\] \[=2{{\log }_{e}}(\sqrt{2}+1)-\sqrt{2}\]