Question

The area bounded by $y=x\left|\sin x\right|andx-axis$ between $x=0,x=2\pi$ is

• A. $2\mathrm{\pi }$
• B. $3\mathrm{\pi }$
• C. $\mathrm{\pi }$
• D. $4\mathrm{\pi }$

Answer 1

The correct option is D

$4\mathrm{\pi }$

Explanation for correct option

Finding the area of the region bounded by the curves

We will bifurcate the limit to $(0,\mathrm{\pi })and(\mathrm{\pi },2\mathrm{\pi })$ for simple calculation

Required area:

$$\begin{gathered} ={\int }_0^{\mathrm{\pi }}x\sin xdx+\left|{\int }_{\mathrm{\pi }}^{2\mathrm{\pi }}x\sin xdx\right| \\ ={\left(-\mathrm{xcosx}+\mathrm{sinx}\right)}_0^{\mathrm{\pi }}+\left|{\left(-\mathrm{xcosx}+\mathrm{sinx}\right)}_{\mathrm{\pi }}^{2\mathrm{\pi }}\right| \\ =(\mathrm{\pi }+0)-(0+0)+\left|(-2\mathrm{\pi }+0)-(\mathrm{\pi }+0)\right| \\ =\mathrm{\pi }+3\mathrm{\pi } \\ =4\mathrm{\pi } \end{gathered}$$

Hence, option (D) is correct.