By David Bressoud @dbressoud
As of 2024, new Launchings columns appear on the third Tuesday of the month.
FDWK refers to Ross Finney, Frank Demana, Bert Waits, and Dan Kennedy, the team of authors who for many years and multiple editions updated Calculus: Graphical, Numerical, Algebraic, one of the most popular textbooks for the College Board’s Advanced Placement Calculus.
Last month I talked about changes that I helped initiate for the opening chapters of the 6th edition, with further changes for the 7th edition that will appear next year. This month I want to explain my own beliefs about how integration should be taught and some of the changes this has influenced within FDWK.
Integration often has not been taught well. Traditionally, integration begins with the Riemann definition of the definite integral, an incredibly and unnecessarily complicated statement that generations of students have learned to ignore. It is then followed by what has become known as the “Fundamental Theorem of Calculus” which students remember, if at all, as the statement that integrals can be found by reversing differentiation. This is not entirely wrong, but when students come to believe that integration is simply the reverse of differentiation, it ceases to be a theorem. Students are then subjected to extensive practice in finding anti-derivatives, reinforcing the notion that this is what integration is all about. Finally, they see applications that are all very mysterious. For most students, it is never really clear why integration provides solutions to these problems.
The fact is that integration models problem of accumulation. Unless students know this, whatever else they learn about integration is useless. The College Board’s Course and Exam Description (CED) for AP Calculus nods in this direction. Its first integration topic is “Exploring Accumulations of Change.” That sounds promising, but I am disappointed that the CED only mentions accumulation as a means of finding areas. It then launches into a detailed analysis of how to find a variety of areas. In FDWK we begin with finding distance knowing velocity, finding snow accumulation knowing rates of snowfall, and how to view the volume of a sphere as an accumulation of volumes of incredibly thin discs. We finish this introductory section with a more unusual use of accumulation to measure cardiac output.
The problem with Riemann’s definition of integration is that he crafted it to explore the question of how discontinuous a function might still be integrable. He showed that it can be very discontinuous, as demonstrated by his example of an integrable function on [0, 1] with a jump discontinuity at every rational number with an even denominator. (See Bressoud, 2007, pp 252–254). But students in the first year of calculus do not need to know this. In fact, the only functions studied in the first year of calculus are piecewise continuous.
An equivalent but more transparent definition of the definite integral was given by Gaston Darboux in 1875. As with Riemann, we begin with an arbitrary partition of the interval over which we are integrating. But we consider two approximating sums now known as the upper and lower Darboux sums. For the upper Darboux sum, we multiply each subinterval by the maximum value of the function on that interval and sum these products. For the lower Darboux sums we use the minimum value. (To be completely rigorous, use the least upper and greatest lower bounds.) I modified the introduction to integration in the 6th edition of FDWK to begin with these Darboux sums as shown in Figure 1.
As explained in the text, this figure shows the “rectangles that extend to the maximum, Mk, and minimum, mk values on each interval. The area shaded with diagonal lines represents the difference between the upper and lower Darboux sums”. If we make the intervals smaller, the upper and lower Darboux sums get closer. If we can make the difference arbitrarily small, then the definite integral exists. In other words, if there is a unique value that is less than or equal to every upper Darboux sum and greater than or equal to every lower Darboux, then the integral exists and equals this value. This is completely equivalent to the Riemann definition. Darboux’s definition is not particularly helpful when questioning which discontinuous functions are integrable. It is wonderfully suited to understanding what we are doing when integrating a piecewise continuous function.
I love to tilt at windmills. One that has consistently defeated me is the usage by modern textbooks of the term Fundamental Theorem of Calculus. The first time this result was called a fundamental theorem was by Paul du-Bois-Reymond in 1880. He called it the Fundamental Theorem of Integral Calculus. That actually was the name that stuck until the 1960s when American calculus textbooks started dropping the adjective “integral.” Even George Thomas in the first edition of his Calculus and Analytic Geometry, published in 1951, called it the Fundamental Theorem of Integral Calculus. I do not know why “integral” was later dropped, but I think it was a serious mistake.
What this theorem accomplishes is to tie together two very different ways of understanding integration: as a limit of sums expressing an accumulation and as reversing differentiation. If students only remember the latter, they are left with useless information. An illustration of how damaging this can be found in Joseph Wagner’s 2017 paper, “Students’ Obstacles to Using Riemann Sum Interpretations of the Definite Integral.” I discussed this in my Launchings column from April, 2018.
For the 5th edition of FDWK, the first on which I worked, I made a point that there are two sides to this fundamental theorem.
“This relationship between accumulation and antidifferentiation is often referred to as The Fundamental Theorem of Calculus. That is a contraction of its original name, The Fundamental Theorem of Integral Calculus. We will refer to this result as the Fundamental Theorem of Calculus, but it is important to remember that this is really a theorem about integration. It says that the definite integral can be understood as accumulation (a limit of Riemann sums) or as antidifferentiation (the change in the value of an antiderivative). As long as we are working with continuous functions, the two ways of thinking about the definite integral are equally valid.
“This connection provides powerful insight in both directions. It says that if we have an accumulation problem, there is an easy way to evaluate it if we can find an antiderivative. It also says that if we need an antiderivative, we can always build one using the accumulation given by the limit of a Riemann sum.” (p. 308 in the 6th edition)
My final column on FDWK, to appear next month, will look at FDWK’s treatment of infinite series, an exemplary introduction to infinite series that I inherited from the earlier authors.
References
Bressoud, D.M. (2007). A Radical Approach to Real Analysis. 2nd edition. Mathematical Association of America.
Bressoud, D.M. (2018). Gaps in Student Understanding of the Fundamental Theorem of In tegral Calculus. Launchings, April, 2018. https://launchings.blogspot.com/2018/04/gaps in-student-understanding-of.html
Darboux, G. (1875). Mémoire sur les fonctions discontinues. Annales scientifiques de
l’École Normale Supèrieure ́ 4, 57–112.
Demana, F., Waits, B., Kennedy, D. Bressoud, & Boardman, M. (2020). Calculus: Graph- ical, Numerical, Algebraic. 6th edition. Pearson.
du Bois-Reymond, P. (1880). Der Beweis des Fundamentalsatzes der Integralrechnung. Mathematische Annalen 16, no. 1. 115–120.
Thomas, G.B. (1951). Calculus and Analytic Geometry. Addison-Wesley.
Wagner, J.F. (2017). Students’ obstacles to using Riemann sum interpretations of the definite integral. International Journal of Research in Undergraduate Mathematics Education, 327–356. https://doi.org/10.1007/s40753-017-0060-7
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. Information about him and his publications can be found at davidbressoud.org