Question

The area bounded by the curve f(x) = x + sin x and its inverse function between the ordinates $x=0andx=2\pi$ is

• A. $4\pi$
• B. $8\pi$
• C. $4$
• D. $8$

Answer 1

The correct option is D $8$

R.E.F image

$f\left(x\right)=x+\sin x$

So area between $f\left(x\right)$ & $f^{-1}\left(x\right)$ is double

the area between $f\left(x\right)\&y=x$

Area (A) = ${\int }_0^{2\pi }(x+\sin x{)}^{dx}{\int }_0^{2\pi }xdx$

$={\int }_0^{2\pi }\sin xdx$

$={\int }_0^{\pi }\sin xdx-{\int }_{\pi }^{2\pi }\sin xdx$

$=[-\cos x{]}_0^{\pi }-[-\cos x{]}_{\pi }^{2\pi }$

$=2+2=4$

Actual Area = $2\times A=8$

Option D is correct.