Question

The area bounded by the curve ƒ(x) = x + sin x and its inverse function between the ordinates x = 0 to x = 2p is-

• A. 4p sq. units
• B. 8p sq. unsits
• C. 4 sq. units
• D. 8 sq. units

Answer 1

Answer: (D)

8 sq. units

Answer 2

Clearly, ƒ: [0, 2p] ® [0, 2p] given by ƒ(x) = x + sin x is a bijection. So, its inverse exists. The graph of ƒ^–1(x) is the mirror image of the graph of ƒ(x) in the line y = x.

\ Required area = Area of the shaded region = 4(Area of the one loop) = 4 $\left[ \int _ { 0 } ^ { \pi } ( x + \sin x ) d x - \int _ { 0 } ^ { \pi } x d x \right]$ = 4 = – 4 [cos p – cos 0] = 8 sq. units. Hence (4) is the correct answer.