Integrating the unit step function \(u\left(t\right)\) results in the unit ramp function \(r\left(t\right)\).
The unit ramp function \(r\left(t\right)\) is zero for negative values of \(t\) and has a unit slope for positive values of \(t\).
\(\boxed{r\left(t\right) = \int_{-\infty}^{t} u\left(\lambda\right)\,d\lambda = t\,u\left(t\right) = \begin{cases}0\,, & t \le 0 \\ t\,, & t \ge 0\end{cases}}\)
In general, a ramp is a function that changes at a constant rate.
For the delayed unit ramp function,
\(\displaystyle r\left(t - t_0\right) = \begin{cases}0\,, & t \le t_0 \\ t - t_0\,, & t \ge t_0\end{cases}\)
For the advanced unit ramp function,
\(\displaystyle r\left(t + t_0\right) = \begin{cases}0\,, & t \le -t_0 \\ t + t_0\,, & t \ge -t_0\end{cases}\)
