Question

Evaluate the following integral:

\[\int\limits_0^{2\pi} \left| \sin x \right| dx\]

 

Answer 1

\[\int_0^{2\pi} \left| \sin x \right| d x\]
\[\text{We know that}, \left| \sin x \right| = \begin{cases} - \sin x &,& \pi \leq x \leq 2\pi\\\sin x&,& 0 < x \leq \pi\end{cases}\]
\[ \therefore I = \int_0^{2\pi} \left| \sin x \right| dx\]
\[ \Rightarrow I = \int_0^\pi \sin x dx + \int_\pi^{2\pi} - \sin x dx\]
\[ \Rightarrow I = - \left[ \cos x \right]_0^\pi + \left[ \cos x \right]_\pi^{2\pi} \]
\[ \Rightarrow I = 1 + 1 + 1 - \left( - 1 \right)\]
\[ \Rightarrow I = 4\]