Definite IntegrationHard
Question
If f(x) = sin x, ∀ x ∈ 
, f(x) + f(π - x) = 2. ∀ x ∈ 
and f(x) = f(2π - x), ∀ x ∈ (π, 2π], then the area enclosed by y = f(x) and x-axis is
, f(x) + f(π - x) = 2. ∀ x ∈ and f(x) = f(2π - x), ∀ x ∈ (π, 2π], then the area enclosed by y = f(x) and x-axis isOptions
A.π
B.2π
C.2
D.4
Solution
f(x) + f(π - x) = 2. ∀ x ∈ 
f(x) = 2 - sin (π - x)
f(x) = 2 - sin x, ∀ x ∈
f(x) = 2 - f(2π - x), ∀ x ∈
f(x) = 2 + sin x, x ∈
f(x) = f(2π - x), ∀ x ∈
f(x) = - sin x, ∀ x ∈
Clearly, from figure required area = 2π
f(x) = 2 - sin (π - x)
f(x) = 2 - sin x, ∀ x ∈
f(x) = 2 - f(2π - x), ∀ x ∈
f(x) = 2 + sin x, x ∈
f(x) = f(2π - x), ∀ x ∈
f(x) = - sin x, ∀ x ∈
Clearly, from figure required area = 2π
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