The concept of accumulation in calculus

The concept of accumulation is central to the idea of integration, and therefore is at the core of understanding many ideas and applications in calculus. On one hand, the idea of accumulation is trivial. You accumulate a quantity by getting more of it. We accumulate injuries as we exercise. We accumulate junk as we grow older. We accumulate wealth by gaining more of it. There are some details to consider, such as whether it makes sense to think of accumulating a negative amount of a quantity, but the main idea is straightforward. On the other hand, the idea of accumulation is anything but straightforward. First, students find it hard to think of something accumulating when they cannot conceptualize the “bits” that accumulate. To understand the idea of accomplished work, for example, as accruing incrementally means that one must think of each momentary total amount of work as the sum of past increments, and of every additional incremental bit of work as being composed of a force applied through a distance. Second, the mathematical idea of an accumulation function, represented as, involves so many moving parts that it is understandable that students have difficulty understanding and employing it. Readers already sophisticated in reasoning about accumulations may find it surprising that many students are challenged to think mathematically about them. The ways in which it is difficult, though, are instructive for a larger set of issues in calculus.