Area of region bounded by y = sin^–1 x, y = cos^–1 x and the x-axis is _____ sq. units

Required Area =

${\int }_0^{1/\sqrt{2}} {\sin }^{-1}\mathrm{xdx}+{\int }_{1/\sqrt{2}}^1 {\cos }^{-1}\mathrm{xdx}$

$\begin{array}{l}\mathrm{Area} \mathrm{ecnlosed} \mathrm{by} \mathrm{the} \mathrm{given} \mathrm{curves} \mathrm{y}={\sin }^{-1}\mathrm{xdx},\mathrm{y}={\cos }^{-1}\mathrm{x} \mathrm{and} \mathrm{x}-\mathrm{axis} \mathrm{is}\\ ={\int }_0^{1/\sqrt{2}} {\sin }^{-1}xdx+{\int }_{1/\sqrt{2}}^1 {\cos }^{-1}xdx\\ ={\left[x {\sin }^{-1}x+\sqrt{1-x^2}\right]}_0^{1/\sqrt{2}}+{\left[x{\cos }^{-1}x-\sqrt{1-x^2}\right]}_{1/\sqrt{2}}^1\\ =\left[\left(\frac{1}{\sqrt{2}}{\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)+\sqrt{1-\frac{1}{2}}\right)-(0+\sqrt{1-0})\right]+\\ \left[\left({\cos }^{-1}\left(1\right)-\sqrt{1-1}\right)-\left(\frac{1}{\sqrt{2}}{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)-\sqrt{1-\frac{1}{2}}\right)\right]\\ =\left[\left(\frac{1}{\sqrt{2}}{\sin }^{-1}\left(\sin \frac{\pi }{4}\right)+\sqrt{\frac{1}{2}}\right)-1\right]+\\ \left[\left({\cos }^{-1}\left(\cos 0\right)\right)-\left(\frac{1}{\sqrt{2}}{\cos }^{-1}\left(\cos \frac{\pi }{4}\right)-\sqrt{\frac{1}{2}}\right)\right]\\ =\frac{\pi }{4\sqrt{2}}+\frac{1}{\sqrt{2}}-1-\frac{\pi }{4\sqrt{2}}+\frac{1}{\sqrt{2}}\\ \mathrm{Required} \mathrm{area}=\sqrt{2}-1\end{array}$