The area (in square units) bounded by the curve $f(x) = x + \sin x$ and its inverse function between the ordinates $x = 0$ to $x = 2π$, is
Answer & explanation
Correct answer: option 4
Clearly, $f : [0, 2 π] ∈ [0, 2 π]$ given by $f(x) = x + \sin x$ is a bijection. So, its inverse exists. The graph of $f^{-1}(x)$ is the mirror image of the graph of f(x) in the line $y = x$.
∴ Required area A is given by
A = 4 (Area of one loop)
$⇒A=4\left[\int\limits_0^π(x+\sin x)dx-\int\limits_0^πx\,dx\right]$
$⇒A=4\int\limits_0^π\sin x\,dx-4\left[\cos x\right]_0^π$
$⇒A=-4[\cos π-\cos 0]=8$ sq.units.