Question

Area of the figure bounded by Y-axis, $y=Si{n}^{-1}x,y=Co{s}^{-1}x$ and the first point of intersection from the origin is

• A. $2\sqrt{2}$
• B. $2\sqrt{2}+1$
• C. $\sqrt{2}-1$
• D. $\sqrt{2}+1$

Answer 1

Answer: (C)

$\sqrt{2}-1$

The curve $y={\sin }^{-1}x$ and $y={\cos }^{-1}x$ intesect at $(\frac{1}{\sqrt{2}},\frac{\pi }{4})$
Thus area $={\int }_0^{\frac{1}{\sqrt{2}}}({\cos }^{-1}x-{\sin }^{-1}x)dx$
$={\left[x{\cos }^{-1}x\right]}_0^{\frac{1}{\sqrt{2}}}-{\int }_0^{\frac{1}{\sqrt{2}}}x.\frac{-1}{\sqrt{1-x^2}}dx+{\left[x{\sin }^{-1}x\right]}_0^{\frac{1}{\sqrt{2}}}+{\int }_0^{\frac{1}{\sqrt{2}}}x.\frac{1}{\sqrt{1-x^2}}dx=\sqrt{2}-1$