The area bounded by the curve f(x) = x + sin x and its inverse function between the ordinates x = 0

Question

The area bounded by the curve f(x) = x + sin x and its inverse function between the ordinates x = 0 and x = 2&#960; is

Accepted Answer

= 1 + cos x ≥ 0, ∀ x
= - sin x
x = nπ are points of inflection, where curve changes its concavity.
For x ∈ (0, π), sin x > 0 ⇒ x + sin x > x.
For x ∈ (π, 2π), sin x < 0 ⇒ x + sin x < x.
Required area = 4A
A =
A = 2
Required area = 4A = 8

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