Question

The area bounded by the curve $y={\sin }^{-1}x$ and the lines $x=0,|y|=\frac{\pi }{2}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

We have to find the area of the shaded region.

Let's first complete the area bounded by the curve

$y={\sin }^{-1}x$ $xϵ[-1,1]$

Since the graph is symmetrical about the origin the area bounded is given by

$A=2{\int }_0^1{\sin }^{-1}xdx$

$=2[{\left|x{\sin }^{-1}x\right|}_0^1-{\int }_0^{1}x\frac{1}{\sqrt{1-x^2}}dx]$

$=2\left[{|\frac{\pi }{2}+\sqrt{1-x^2}|}_0^1\right]$

$=\pi -2$

$\therefore$ Area of the shaded region is

$=|Ar\left(OABC\right)+Ar\left(ODEF\right)-A|$

$=|\frac{\pi }{2}\times 1+\frac{\pi }{2}\times 1-\pi +2|$

$=\left|2\right|$

$=2$ units

Hence, this is the answer.