Consider two functions `f (x) ={[x] , -2 leq x leq -1 and |x|+1 , -1 lt x leq 2 and g(x)={[x], -pi leq x lt 0 and sin x and 0 leq x leq pi`, where [.] denotes the greatest integer function.

Options

A. `(sqrt(3))/(4)+(pi)/(6)` sq. units
B. `(sqrt(3))/(2)+(pi)/(6)` sq. units
C. `(sqrt(3))/(4)-(pi)/(6)` sq. units
D. `(sqrt(3))/(2)-(pi)/(6)` sq. units

Answer

A

Explanation

`A=2overset(1//2)underset(0)intsqrt(1-x^(2))dx`
`=2[(x)/(2)sqrt(1-x^(2)):|_(0)^((1)/(2))+(1)/(2) sin^(-1)x:|_(0)^((1)/(2))]`
`= (sqrt(3))/(4)+(p)/(6)` sq. unit.