Question

The area bounded by the curve `f(x)= x + sinx` and its inverse function between the ordinates `x= 0` and `x=2pi` is (A) `4pi` square units (B) `8pi` square units (C) 4 square units (D) 8 square units
A. `4pi`
B. `8pi`
C. 4
D. 8

Answer 1

Correct Answer - d
Clearly, `f:[0,2pi]in[0,2pi]"given by f"(x)=x+sinx` is 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.`

`therefore` Required area A is given by
A=4 (Area of one loop)
`impliesA=4[underset(0)overset(pi)(int)(x+sinx)dx-underset(0)overset(pi)(int)x dy]`
`impliesA=4underset(0)overset(pi)(int)sinxdx=-4[cosx]_(0)^(pi)`
`impliesA=-4[cospi-cos 0]=8` sq. units.