The area bounded by the graph of `y=f(x), f(x) gt0` on [0,a] and x-axis is `(a^(2))/(2)+(a)/(2) sin a +(pi)/(2) cos a ` then find the value of `f((pi)/(2))`.

Options

A. `(1)/(2)`
B. `(a)/(2)`
C. `(a^(2))/(2)`
D. `(pi)/(2)`

Answer

A

Explanation

According to the given condition,
Area of curve `=int_(0)^(a)f(x)dx`
`rArr(a^(2))/(2)+(a)/(2)"sin a"+(pi)/(2)cosa=int_(0)^(a)f(x)dx`
On differentiating both sides w.r.t. a, we get
`a+(1)/(2)sina+(a)/(2)cosa-(pi)/(2)sina=f(a)`
`therefore f((pi)/(2))=(pi)/(2)+(1)/(2)sin.(pi)/(2)+(pi)/(4)cos.(pi)/(2)-(pi)/(2)sin.(pi)/(2)`
`rArrf((pi)/(2))=(pi)/(2)+(1)/(2)-(pi)/(2)`
`rArrf((pi)/(2))=(1)/(2)`