Calculate total variation $V_0^{2\pi}(f) $

Question

I am reviewing an analysis book and this exercise comes, in another example using the piecewise function many steps are skipped. Will you help me understand how to make my partitions with this example please.

$$f(x)=\begin{cases}1&\text{ for }x=0 \\ x&\text{ for }x\in \left ( 0,\pi \right ]\\ \sin x&\text{ for }x\in \left ( \pi,2\pi \right ]\end{cases}$$

Answer 1

$$V_0^{2\pi} = 1+\int_0^{\pi} |(x)' |dx +\pi +\int_{\pi}^{2\pi} |(\sin x )' |dx$$