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Let f(x) be a continuous function with f(x) ≥ 0. The area enclosed by the graph y = f(x), the x-axis, and the vertical lines x = π/4 and x = β, where β > π/4, is given by βsinβ + (π/4)cosβ + √(2) β. Then the value of f(π/2) is:

1. (π/4 + √(2) - 1)
2. (π/4 - √(2) + 1)
3. (1 - π/4 - √(2))
4. (1 - π/4 + √(2))

Correct answer: (1 - π/4 + √(2))

Solution

The correct option is derived from evaluating the area under the curve f(x) from x = rac{4} to x = eta, using the given area formula. By substituting eta = rac{2} into the area expression and simplifying, we find that f( rac{2}) equals 1 - rac{4} + ext{sqrt}(2), confirming option D as the correct answer.

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