The area bounded by the curves `y=cos^(-1)x,y =sin^(-1) x and y=-pix^(3),` where `-1lex le1`,is

Options

A. `(3pi)/(2)+1-sqrt2` sq. units
B. `(3pi)/(4)+1+sqrt2` sq. units
C. `(3pi)/(4)+2-sqrt2` sq. units
D. `(3pi)/(4)+1-sqrt2` sq. units

Answer

D

Explanation

Graph of the function is as shown in the following figure:

From the figure, bounded area
`A=|int_(0)^(pi)((-y)/(pi))^(1//3)dy|-|int_(pi//2)^(pi)cos ydy|+|int_(pi//4)^(pi//2)cos ydy|+|int_(0)^(pi//4)sinydy|`
`=|(-3y^(4//3))/(4pi^(1//3))|_(0)^(pi)-|siny|_(pi//2)^(pi)+|siny|_(pi//4)^(pi//2)+|-cosy|_(0)^(pi//4)`
`=|-(3)/(4)pi|-|-1|+|1-(1)/(sqrt2)|+|-(1)/(sqrt2)+1|`
`=(3)/(4)pi-1+1-(1)/(sqrt2)+1-(1)/(sqrt2)`
`=(3pi)/(4)+1-sqrt2`