Question

The area ( in square units ) bounded by the curve $y=\left|c o s^{- 1} \left(sin x\right)\right|+\left|\frac{\pi }{2} - c o s^{- 1} \left(cos ⁡ x\right)\right|$ and the $x$ -axis, where $\frac{\pi }{2}\leq x\leq \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)

$y=\left|\frac{\pi }{2} - s i n^{- 1} sin x\right|+\left|\frac{\pi }{2} - c o s^{- 1} cos ⁡ x\right|$
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.