Question

Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), x-axis and the ordinates x = `π/4` and x = `β > π/4` is `(βsinβ + π/4 cos β + sqrt(2)β)`. Then `f(π/2)` is ______.

विकल्प

• `(π/4 + sqrt(2) - 1)`

• `(π/4 - sqrt(2) + 1)`

• `(1 - π/4 - sqrt(2))`

• `(1 - π/4 + sqrt(2))`

Answer 1

Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), x-axis and the ordinates x = `π/4` and x = `β > π/4` is `(βsinβ + π/4 cos β + sqrt(2)β)`. Then `f(π/2)` is `underlinebb((1 - π/4 + sqrt(2))`.

Explanation:

From given condition

`int_(π//4)^β f(x)dx = βsinβ + π/4 cos β + sqrt(2)β`

Differentiating w.r.t β, we get

f(β) = `β cos β + sin β - π/4 sin β + sqrt(2)`

`f(π/2) = β.0 + (1 - π/4) sin  π/2 + sqrt(2)`

`f(π/2) = 1 - π/4 + sqrt(2)`