How do I go from an infinite amount of instantaneous rates to an average rate?

Question

I have a function that gives me rate at a point (the derivative) and I have a value for average rate that was experimentally calculated. How would I go about creating an expression to solve for a theoretical average rate. My intuition tells me to integrate the rate function and divide by the total number of things I added, but that makes me divide by infinity and average rate = 0, which is not correct. If it helps, I am dealing with volume flow rate, where dV/dt is volume flow rate and I need to find average over some time t for a physics experiment. Thanks

Answer 1

The average rate of change is $\frac{\Delta V}{\Delta t}$. The instantaneous rate of change is $\frac{dV}{dt}$ is the instantaneous rate of change.

The total change is the integral of the derivative: $\Delta V = \int_{t_1}^{t_2} \frac{dV}{dt}dt$. That also tells us that $\Delta t=t_2-t_1$

So $\frac{\Delta V}{\Delta t }=\frac{1}{t_2-t_1}\int_{t_1}^{t_2} \frac{dV}{dt} dt$

More generally $\overline F = \frac{1}{x_2-x_1}\int_{x_1}^{x_2} F(x) dx$. This integral expression defines an average of the function. It is the Mean Value Theorem for integrals.