Ok, we all know that definite integration of velocity functions, dy/dx, or f' gives you displacement/distance of the function (depending on the situation).

However, I have encountered situations where I used definite integration of differentiated functions to get the "quantity" of something. For example, suppose a graph shows the rate of leaves being produced over a season; you would use definite integration of the rate function in a given interval to calculate the sum of leaves produced. Nothing to do with displacement/distance!

This example brings my point. Why can definite integration be used in two completely different ways? I'm just worried that I may be missing something. Maybe standard velocity/dy&dx/rate/f' equations are derived in terms of the x-axis and the other rate equation forms (designed for quantity problems) are actually in terms of "t" (time)? I just don't know. I'd appreciate it so much if someone could help me out. Thanks!!!