For \(T_2\), since, it is \(T_2>\Delta t\), we have instead to take the difference of the s-function between the two instant \(T_2=t_2-t_{in}\) and \(T_2-\Delta t\ =\ t_2-t_{in}-\Delta t\).
Please observe that in the previous derivation Figure 3 can be misleading. In fact the integration in equation \ref{eq1} is over \(t_{in}\) while Figure \ref{900583} represent the IUH vs clock time \(t\). Therefore, in principle, our integral is not the marron area below the IUH (it is just in a special case, which is actually the one we are dealing with) and the change of variable we did (exchanging \(t_{in\ }\)with the travel time \(T\)) hides this fact. A right representation of the integration is, instead shown in Figure X below.