Last block in Differential Calculus, Proofniks learned how to take a snapshot of how a function is changing at a given point. Now, in Integral Calculus, they are learning how to reassemble those snapshots into a view of the whole.

The two subjects are opposites in one way, yet they are intimately related by the Fundamental Theorem of Calculus, a beautiful and deep result.

In a rush to get to problem-solving algorithms, many calculus classes present the Fundamental Theorem very early in the development of the integral. This does make calculations “easy”, but it can deprive students of a supple understanding of the definite integral, making it hard for them to apply the concept later—especially outside the math classroom, such as in physics. Thus, I chose to delay the Fundamental Theorem and emphasize the nature of the definite integral as a “high-resolution” version of the Riemann sum, a sum of many small incremental products. Such sums can appear wherever we consider the product of two changing quantities, like velocity and time, length and area, or force and distance.

As students have added mathematical techniques to their repertoire, each has been illustrated through examples requiring them to interpret results in a physical context, such as the water flowing in and out of a reservoir or the energy needed to heat a building. In parallel, wherever possible, our analysis has been supported by graphs and geometric reasoning. This approach has had the happy side effect of engaging students who already know some of the rote computations of integral calculus, but who can still be challenged by deeper conceptual questions—some of which have required them to abandon preconceptions!

We are starting each unit of the class with a warm-up problem that requires imagination, not learned methods. Here is a typical example: A rising curve is cut by two horizontal lines as shown. We are permitted to draw a vertical line of our choice through any point along the curve, creating two bounded areas (shaded). How should we position the vertical line so as to minimize the total of the two areas?


No formula is given to describe the curve; the answer to the problem must be generally applicable. The problem can be solved with the Fundamental Theorem of Calculus (and some did solve it that way), but the most elegant solution—which was also found by many—is one that could be explained to a person with no calculus background. Can you figure it out?

As an aside, this problem has an unsavory history. It is from a set of problems collected by Tanya Khovanova, which required more ingenuity to solve than standard test questions and were used in the Soviet Union to discriminate against Jewish applicants to universities. Dr. Khovanova collected these problems for both their historical and their mathematical value. As she wrote in her preface, “Now, after thirty years, these problems seem easier. Mostly, this is because the ideas of how to solve these problems have spread and are now a part of the standard set of ideas”—or, at least, the standard set of ideas about math education at places like math circles and Proof School. The fact that half my class could figure out the warm-up on their own should not trivialize the effectiveness of the discriminatory impact Dr. Khovanova recounts in her paper, which you can read here. And I can attest that such problems are still alien and frightening to students who have not developed a rich inner sense of geometry!

Complementary to the warm-ups, we finish each class with a “check-up” of differential calculus skills. These check-ups consist of AP test questions, so they should be especially helpful to students who are planning to take the AP. But they are also a good reinforcement for everyone, and I have been able to address some of the weaknesses that have been exposed. On our second quiz, each student got a custom page with an opportunity to redo two problems missed on earlier check-ups. When a large number of absences due to the Bay Area Math Olympiad disrupted my planned lesson, we took advantage of the downtime with an impromptu review of differential calculus topics suggested by students and by their teacher from last block.

To accommodate different learning paces while still pushing everyone to their limits, I am using a series of supplemental handouts—optional explorations of everything from the proto-integral calculus of Archimedes to “pathological” integrals to the surprising history of the logarithm. And since it’s equally important to keep things light and lively, students have enjoyed finding groanworthy calculus puns secreted into the handouts, including a mock check-up that required them to assemble a punchline from separate pieces given to each of them. Just because we’re learning calculus doesn’t mean we can’t have fun!

— Austin Shapiro