I am dealing with a step function S(t). The true functional form is not given or unknown, but what is known is that S(t) takes a different value at each time point t like this below.
t S(t) W(t)
0 1 48.75
1 0.996 48.6
4 0.993 48.0
6 0.9989 48.46
7 0.9986 48.32
14 0.9982 48.18
I know that $\int\limits_0^ts(u)\,\mathrm du= \sum_{i=2}^{i}(t_i - t_{i-1})*S(t_i)$ , Riemann sum concept.
How do I solve $\int\limits_0^t us(u)\,\mathrm du=??$ is it similar to the above Riemann sum concept, $\int\limits_0^t us(u)\,\mathrm du=\sum_{i=2}^{i}(t_i - t_{i-1})*t_i*S(t_i)$
Any advise or pointers on solving this integral is much appreciated.