unknown Wrote:yeah but the idea of v-bar, average time for an entire journey is not derived from calculus, neithe was anything in my post. (the one that I didnt make a stupid mistake in that is)

The fact that you use an average velocity over an infitesimal period of time to approximate a function is calculus. When you use the "average" velocity formula in a numerical integrator, it becomes trapezoidal integration.