Using Accumulation Functions and Definite Integrals in Applied Contexts

A rate tells you how fast something is happening at each instant, like gallons per hour or people per year. If you slice the time interval into tiny pieces, the rate times each tiny piece of time gives a tiny amount of stuff. Add up all those tiny amounts and you get the total change, which is exactly what a definite integral computes: the area under the rate curve. That is why integral of R(t) dt from a to b gives you the total gallons pumped, oil leaked, or water collected between times a and b, not a rate at all.

Units make this concrete. If R(t) is in gallons per hour and dt is in hours, multiplying them cancels the hours, leaving gallons. That is why the accumulated total always has different units than the rate itself: the integral strips away the per-time part. This is the same reasoning behind position being the integral of velocity, or total cost being the integral of marginal cost. When the input variable is not time, the idea still works. If R(x) is measured per person, integrating over a range of people gives a total amount, not a per-person rate, because dx carries the units of people and cancels the per-person part of R.

This topic sits right next to using accumulation functions to build F(x) = integral from a to x of f(t) dt, and connects forward to average value, where you divide a total by the length of the interval to recover a rate-like quantity. Keep the picture of area under a rate curve in mind and both ideas stay straightforward.