The entire story to my original question is based on creating a reliability report. The resistor I stated in my original question needs to have a maximum power dissipation value based on some length of time. It's a 3W resistor, but we need to place a value for a short (instantaneous) value.
So in other words, after a long time, the resistor dissipates nearly zero watts. In our report, we'd state it's a 3W, and, after a length of time, it dissipates approximately 0W. On another line, we need a specification for instantaneous wattage, and then how much the resistor dissipates.
This is why I thought taking the integral from 0s to something like 1ms would give the total power.
As an example, if the resistor can handle a peak of 1000W for 1ms (although I haven't seen such a spec), then I'd imagine taking the integral from 0s to 1ms would be the correct way to provide an "accurate" number.