Question

Find the area bounded by $y=x+sinx$ and its inverse between $x=0$ and $x=2\pi$

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (D)

$8$

graph of $x+\sin x$ and its inverse is shown in the figure which is symmetric with $y=x$

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

from the figure it is clear that $A={\int }_0^{2\pi }(f\left(x\right)-f^{-1}\left(x\right))dx=4{\int }_0^{\pi }(f\left(x\right)-x)dx=4{\int }_0^{\pi }(x+\sin x-x)dx=8$