Draw the graph of `y=sin^(-1)|sin x|and y=(sin^(-1)|sinx|)^(2),0lexle2pi`

Options

A. `(1)/(3)+(pi^(2))/(4)` sq. units
B. `(1)/(6)+(pi^(3))/(8)` sq. units
C. 2 sq. units
D. None of these

Answer

D

Explanation

`y=sin^(-1)|sin x|={{:(x",",0lexlt(pi)/(2)),(pi-x",",(pi)/(2)lexltpi),(pi-x",",pilexlt(3pi)/(2)),(2pi-x",",(3pi)/(2)lexlt2pi):}`
`y=(sin^(-1)|sin x|)^(2)={{:(x^(2)",",0lexlt(pi)/(2)),((pi-x)^(2)",",(pi)/(2)lexltpi),((x-pi)^(2)",",pilexlt(3pi)/(2)),((2pi-x)^(2)",",(3pi)/(2)lexlt2pi):}`
The required area A is shown as shaded region in the figure.

`rArr" "4overset(1)underset(0)int(x-x^(1))dx+4overset(pi//2)underset(1)int(x^(2)-x)dx`
`=(4)/(3)+pi^(2)[(pi-3)/(6)]` sq. units.