Question

Evaluate the following integrals:
$\underset{0}{\overset{2\mathrm{\pi }}{\int }}\left|\sin x\right|dx$

Answer 1

$$\begin{gathered} {\int }_0^{2\mathrm{\pi }}\left|\sin x\right|\mathit{d}x \\ \mathrm{We}\mathrm{know}\mathrm{that},\left|\sin x\right|=\left\{\begin{array}{l}-\sin x,\mathrm{\pi }\le x\le 2\mathrm{\pi }\\ \sin x,0<x\le \mathrm{\pi }\end{array}\right. \\ \therefore I={\int }_0^{2\mathrm{\pi }}\left|\sin x\right|\mathit{d}x \\ \Rightarrow I={\int }_0^{\mathrm{\pi }}\sin xdx+{\int }_{\mathrm{\pi }}^{2\mathrm{\pi }}-\sin xdx \\ \Rightarrow I=-{\left[\cos x\right]}_0^{\mathrm{\pi }}+{\left[\cos x\right]}_{\mathrm{\pi }}^{2\mathrm{\pi }} \\ \Rightarrow I=1+1+1-\left(-1\right) \\ \Rightarrow I=4 \end{gathered}$$