The area bounded by the curve `y=|cos^(-1)(sinx)|-|sin^(-1)(cosx)|` and axis from `(3pi)/(2)lex le 2pi`

Options

A. `pi^(2)` sq. units
B. `pi^(2)//4` sq. units
C. `pi^(2)//2` sq. units
D. none of these

Answer

B

Explanation

We have `y=|cos^(-1)(sinx)|-|sin^(-1)(cosx)`
`=|(pi)/(2)-sin^(-1)(sinx)|-|(pi)/(2)-cos^(-1)(cosx)|`
`=|(pi)/(2)-(x-2pi)|-|(pi)/(2)-(2pi-x)|`
`=|(5pi)/(2)-x|-|x-(3pi)/(2)|`
`=((5pi)/(2)-x)-(x-(3pi)/(2))`
`=4pi-2x`
The graph of the function is as shown in the following figure.

From the figure, Requried area
= Area of shaded triangle in figure
`=(1)/(2)xx((pi)/(2))xxpi=(pi^(2))/(4)`