One of the main uses of the definite integral is summed up (pun intended) in the idea of accumulation. When you integrate a rate of change you get the (net) amount of change. This important idea is often treated very lightly, if at all, in textbooks.
Here are a series of past posts that use, explain, and illustrate that concept.
Accumulation – Need an Amount? The Fundamental Theorem of Calculus says that the integral of a rate of change (a derivative) is the net amount of change. This post shows how that works in practice.
AP Accumulation Questions and Good Question 7 – 2009 AB 3 the “Mighty Cable Company” show how accumulation is tested on the AP Calculus exams. The “Mighty Cable Company” question is a particularly good and difficult example.
The next two posts show how to use the concept of accumulation to analyze a function and its graph without reference to the derivative. The graphical idea of a Riemann sum rectangle moving across the interval of integration makes the features of function much more intuitive than the common approach. You will not find these ideas in textbooks. Nevertheless, a lesson on this idea may help your students.
Graphing with Accumulation 1 explains how to analyze the derivative to determine when a function is increasing or decreasing and finding the locations of extreme values. By thinking of the individual Rieman sum rectangles moving across the interval the features of the function are easy to see and easier to remember. Once understood, this method will help students with their graph analysis work.
Graphing with Accumulation 2 continues the idea of using accumulation to determine information about the concavity of a function.