$\text{ Evaluate }{\int }_0^{2\pi } [\sin x]dx,\text{ where [.] denotes the greatest integer function. }$

$y=\sin x\text{ is a non-monotonic function in }[0,2\pi ]\text{ . }$

$\text{ Hence, draw the graph of }f\left(x\right)=[\sin x]\text{ . }$

$\text{ From the above graph we have, }$

$\begin{array}{c}{\int }_0^{2\pi } [\sin x]dx=\text{ Algebraic area of the shaded region }\\ \begin{array}{rl} & =\left(\pi \right)(-1) (l\times b)\\ & =-\pi \end{array}\end{array}$