First slide

Area of bounded Regions

Question

The area bounded by the curve f(x) = x + sin x and its inverse function between the ordinates x= 0 to x = $2 π$ , is

Difficult

A

4 $π$ sq units
B

8 $π$ sq units
C

4 sq units
D

8 sq units

Solution

Clearly, $f : [0, 2 π] \to [0, 2 π]$ 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 = 4 (Area of one loop)
$\begin{array}{l} = 4 [\int_{0}^{π} (x + \sin x) dx - \int_{0}^{π} xdx] \\ = 4 \int_{0}^{π} \sin xdx \\ = - 4 [\cos x]_{0}^{π} \\ = - 4 [\cos π - \cos 0] \\ = 8 sq units \end{array}$

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