Question

The area bounded by the curve $f\left(x\right)=x+\sin x$ and its inverse between the ordinates $x=0$ to $x=2\pi$

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

Answer 1

The correct option is B $8$ sq.unit

Graph of $f^{-1}\left(x\right)$ is the mirror image of $f\left(x\right)$ about the line $y=x$
Required Area$=4A$ ......................$\left(1\right)$
where $A={\int }_0^{\pi }f\left(x\right)dx-\frac{1}{2}\pi .\pi$
$={\int }_0^{\pi }(x+\sin x)dx-\frac{{\pi }^2}{2}$
$={[\frac{x^2}{2}-\cos x]}_0^{\pi }-\frac{{\pi }^2}{2}$
$=[\frac{{\pi }^2}{2}-(-1)]-(0-1)-\frac{{\pi }^2}{2}$
$=2$sq.unit
Required Area$=4\times 2=8$sq.unit