Question

Find the area bounded by $y={\cos }^{-1}x,y={\sin }^{-1}x$ and $y-$ axis.

Answer 1

$\begin{array}{c}y={\cos }^{-1}x,y={\sin }^{-1}xandy-axis\\ \Rightarrow 2{\cos }^{-1}x=\frac{\pi }{2},{\cos }^{-1}x=\frac{\pi }{4},x=\frac{1}{2}\\ \Rightarrow requiredarea={\int }_0^{\pi /2}socydy+{\int }_0^{\pi /4}\sin ydy\\ \Rightarrow Area={\left[\sin y\right]}_0^{\pi /4}+{\left[\cos y\right]}_0^{\pi /4}\\ \Rightarrow Requiredarea=1+\left(\frac{1}{\sqrt{2}}-1\right)=2-\frac{1}{\sqrt{2}}\end{array}$