There is a long-standing tradition of defining integrals as being area and introducing accumulation as an application of this core mathematical idea. What happens if, instead, we define integrals as being accumulation and then link this idea to area? For example, I use this activity to introduce Riemann sums by first having students estimate the total energy released by a supernova over a period of time, and then leading them represent their work in terms of area.
This approach has a clear benefit to student buy-in: students are likely to find accumulation problems more intrinsically interesting than areas. It also has a benefit in terms of giving students some intuition for integrals. For example, in the graphs above, students find it intuitive that the volume graph is steep when the flow rate graph has a large value. There is also a strong historical case for foregrounding accumulation: Newton was certainly more interested in motion than in area per se, and Babylonian astronomers tracking the motion of Jupiter were among the very earliest to use Riemann sums.1
But you may be wondering: is this rigorous? Shouldn’t integrals be defined in terms of a mathematically clear idea like area, rather than a more ill-defined one like accumulation? This post focuses on the theoretical benefits of foregrounding accumulation. I would argue that foregrounding accumulation actually leads to a more theoretically rich (and more rigorous) framework for understanding Riemann integrals.
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First, let us be clear about one thing: Riemann integrals are directional: we don’t integrate between a and b, but rather from a to b. This idea takes on heightened importance in multivariable calculus, where line and surface integrals require integration along oriented manifolds. This idea of orientation—which often gets hidden by discussions of signed area—becomes intuitive and meaningful if integrals are introduced first through accumulation and then through area.
The concept of area is not oriented: surely
$$
\int_a^b f(x)dx \hspace{5mm}\text{and} \hspace{3mm} \int_b^a f(x)dx
$$
both represent the same area. So why is one positive and one negative? Because Riemann integrals, unlike areas, are oriented. We justify this using signed area, but signed area exists to impose orientation on an idea that is not naturally oriented. Leaning too heavily on signed area masks the fact that orientation (the idea that you go from one bound to another) is not some incidental thing that comes up when you reverse the bounds: it is absolutely essential to the nature of a Riemann integral. For example, the Fundamental Theorem would not be true for non-oriented integrals.
Accumulation, unlike area, is naturally oriented: the accumulated change from a to b is clearly the negative of the accumulated change from b to a. In the context of the graphs below, it is easy to see that the integrals $$
\int_{118}^{189} v^\prime(t) {\rm d}t \hspace{5mm}\text{and} \int_{189}^{118} v^\prime(t) {\rm d}t
$$ have opposite meanings: one is the change in volume going from day 118 to 180, the other from day 180 to 118. Since the net flow rate is positive in this period, the lake gains volume moving forward in time. But that means there was less water in the past–hence the integral on the right must be negative. And this is perfectly clear looking at the antiderivative graph: Moving to the right in this period you go up, moving to the left you go down. Signed area is a natural consequence of finding an accumulated change in volume, from one time to another.
In terms of Riemann sums, signed area has its roots in the idea that \(
\scriptsize\Delta x\) has a sign: it is positive if \(\scriptsize x\) incrementally goes from a smaller a to a larger b, and negative if it goes from a larger a to a smaller b. And this makes total sense if we think about accumulating from one point to another: we must use either positive or negative \(\scriptsize \Delta x\) depending on whether we want to incrementally increase \(\scriptsize x\) or decrease it. But thinking of \(
\scriptsize\Delta x\) as being signed makes much less sense if we see integrals as being fundamentally about area. If our primary goal was to compute area, why would we ever want the width of the rectangle to be negative?
Indeed, books that make area the defining idea for integrals often do not include the sign of \(
\scriptsize\Delta x\) in the definition of the definite integral. A typical approach is to define “the definite integral” as always having a positive orientation, then to give an entirely separate definition for an integral with reversed bounds. It is as if there are two types of “definite integral:” one that is described by the stated definition, and some other type in which the bounds can be reversed as long as the sign is changed. But this is a serious mischaracterization of integration: The sign of \(\scriptsize \Delta x\) is an absolutely essential part of the Riemann integral.
It is actually quite easy to write the definition of a definite integral so that \(
\scriptsize\Delta x\) is signed. Most definitions of the definite integral involve dividing the interval \(
\scriptsize [a,b]\) using \(
\scriptsize (x_1, x_2,…,x_n) \), with \(
\scriptsize a=x_1< x_2<…<x_n=b\), specifically requiring that the \(
\scriptsize x_j\)’s are increasing. But if we simply say that \(
\scriptsize (a=x_1, x_2,…,x_n=b) \) is an ordered list going from a to b (either increasing or decreasing), then \(\scriptsize \Delta x=(x_{j+1}-x_j)\) is automatically signed correctly: positive if \(
\scriptsize a<b\) and negative if \(
\scriptsize a>b\). Unless you specifically go out of your way to say that \(
\scriptsize a>b\) and that the \(
\scriptsize x_j\)’s are increasing, then signed area is just a natural part of the definition, not something that requires any separate consideration.
So why doesn’t everyone define the integral this way? Why would almost all books give a separate definition of integration for reversed bounds? Because they start out by telling students that integrals are all about area, and having made this claim they are forced to treat \(
\scriptsize \Delta x\) as by definition positive, which leads them to give an inaccurate definition of the integral and subsequently patch it up by addressing reversed bounds a few pages later. All these problems go away if we are just honest with our students from the start: integrals are always taken over an oriented interval, and this sense of orientation is absolutely essential to the accumulation problems they are meant to solve.
To be clear, I’m not opposed to teaching signed area: it is an essential concept and important for solving accumulation problems, among other things. I’m just saying that we should start by talking about accumulation and let signed area come as a natural result of this work. I’m proposing that we treat signed area as a (very) useful tool, not as an end unto itself.
Making accumulation the defining idea for an integral helps students immediately and intuitively see orientation as a key part of an integral’s nature. It encourages us to define integrals as being oriented, so that the sign of an integral follows naturally from the underlying concept. It also clarifies the importance of integrating from a to b, and it teaches students to expect that integrals in future classes will be taken over \textit{oriented} manifolds.
- Matthieu Ossendrijver. “Ancient Bablynoian Astronomers Calculated Jupiter’s Position from the Area Under a Time-Velocity Graph.” In: Science (2016), p. 482 ↩︎