Question

The area ( in square units ) bounded by the curve $y=\left|co{s}^{-1}\left(sinx\right)\right|+\left|\frac{\pi }{2}-co{s}^{-1}\left(cosx\right)\right|$ and the $x$ -axis, where $\frac{\pi }{2}\le x\le \pi$ , is equal to

• A. ${\pi }^2$
• B. $\frac{{\pi }^2}{2}$
• C. $\frac{{\pi }^2}{8}$
• D. $\frac{{\pi }^2}{4}$

Answer 1

Answer: (D)

$y=\left|\frac{\pi }{2}-si{n}^{-1}sinx\right|+\left|\frac{\pi }{2}-co{s}^{-1}cosx\right|$
For $x\in \left[\frac{\pi }{2},\pi \right]$
$y=\left|\frac{\pi }{2}-\left(\pi -x\right)\right|+\left|\frac{\pi }{2}-x\right|$
$y=\left|x-\frac{\pi }{2}\right|+\left|\frac{\pi }{2}-x\right|$
$y=x-\frac{\pi }{2}+x-\frac{\pi }{2}=2x-\pi$

Area $=\frac{1}{2}\times \frac{\pi }{2}\times \pi$ square units.
$=\frac{{\pi }^2}{4}$ square units.