David M. Bressoud May, 2009
It is an eye-opening experience to question students who have successfully completed the first semester of Calculus and ask them to state the Fundamental Theorem of Calculus (FTC) and to explain why it is fundamental. The best students will remember that this theorem asserts that integration and differentiation are inverse processes. Some will remember the part that asserts that if f is the derivative of F, then the definite integral of f over [a,b] is F(b) – F(a). But even those who remember what the FTC says are stymied at explaining its importance. The problem is that, for most students, their working definition—and frequently their only definition—of integration is anti-differentiation. I have had students explain that this theorem is important because it defines integration.
That, of course, points to a basic problem. I know of no modern calculus textbook that defines the definite integral in any way other than as a limit of Riemann sums. The evaluation part of the FTC, that part of the FTC that most students do remember, is fundamental precisely because the definite integral is defined as a limit of Riemann sums. The FTC provides an alternate and usually much easier route to the evaluation of the integral. But most students never internalize the limit definition. For them, the formative problems of integral calculus are those that involve finding anti-derivatives and evaluating definite integrals through the use of anti-derivatives. Most students can survive calculus, even do quite well in it, with no definition for the integral other than anti-differentiation. Applications to areas, volumes, and accumulations of rates of change can be performed not because students understand why the definite integral is applicable to these problems, but because they have learned how to apply it. The fact that most intructors are reluctant to ask students to solve an unfamiliar problem that first must be cast as a limit of Riemann sums is evidence that few believe that their students really understand the limit definition.
Some of the blame for failing to understand integration as anything other than anti-differentiation has been laid at the feet of the ubiquitous presence of calculus in the high school curriculum. Students know in advance that the painful, cumbersome formulas for direct evaluation of limits of Riemann sums will be swept aside as soon as integration is connected to differentiation. Many instructors—I was once among them—believe that if only we could return to those days when students arrived in college totally ignorant of calculus, then students would believe us when we define the integral as a limit and be suitably awed by the emergence of the FTC.
History suggests otherwise. Riemann's definition of the integral did not appear until 1854 and was not published until 1868. From the emergence of calculus in the late 17th century until Cauchy's attempts in the 1820s to set analysis upon a rigorous foundation, integration was defined as anti-differentiation. Long after Cauchy, the definite integral continued to be defined exactly as our students would define it today, as a difference of values of an anti-derivative, the values taken at the endpoints of the interval .This is how Lacroix defined it in his Traité élémentaire de Calcul Différential et de Calcul Intégral  of 1802, the textbook that dominated calculus instruction in both French and English in the first half of the 19th century. It is how W. A. Granville would define it in the standard American text, Elements of the Differential and Integral Calculus , from its first appearance in 1904 through its last edition of 1957. Until the 1950s, most textbook writers chose to define the definite integral in terms of the anti-derivative.
Beginning with his second edition of 1911, Granville did state a fundamental theorem of calculus. For him it was the statement that the definite integral could be used to evaluate the limit of a Riemann sum evaluated at the midpoints of the intervals. This distinction takes us to the heart of what the FTC really says. The FTC is telling us about integration, that there are two ways of defining the definite integral: in terms of anti-derivatives or as a limit of a Riemann sum. With suitable restrictions, the simplest being that the function to be integrated is continuous over the interval, they are equivalent. When the FTC first appeared in print in its modern form, in an appendix to an1876 paper of Paul du Bois-Reymond , it was descibed as the fundamental theorem of integral calculus (my emphasis), a name that was maintained by Hobson when he popularized it in English in 1907 , and that Granville also used. This really is a theorem about integration.
We do our students a disservice if we do not make this clear. This does not require defining the definite integral other than as a limit of Riemann sums (but see note ) or changing the statement of either part of the FTC. But it does require that we modify how we introduce and explain this theorem. The Evaluation Part of the FTC says that if we have an antiderivative for a continuous function f, then the limit of the Riemann sums of f over the interval [a,b] can be evaluated in terms of the antiderivative of f. The Antiderivative Part of the FTC says that for any continuous function f, the function defined as the limit of the Riemann sums of f with variable upper limit x gives us an antiderivative. In other words, so long as the function to be integrated is continuous, anti-differentiation and limits of Riemann sums give us the same answers.
Surely this is the fundamental insight that we credit to Newton and Leibniz. This is where the real power of calculus arises: that a problem that naturally gives rise to a limit of Riemann sums can be evaluated using antiderivatives, but also that antiderivatives can always be generated from limits of Riemann sums.
Of course, it is never enough just to tell this to students. They have to discover the power of this equivalence for themselves. To do so requires giving them a succession of ever more sophisticated problems for which the students must determine how to translate the problem into the evaluation of a limit of Riemann sums. They are led to see the usefulness of the Evaluation Part of the FTC as a quick and easy step that enables them to find an exact solution (see note ). To understand the power of the Anti-Derivative Part of the FTC, they need to wrestle with problems that seem to call for an antiderivative, but for which an answer can be determined to whatever degre of accuracy is required through the use of an equivalent Riemann sum (see note ).
We need to restore the recognition that the FTC is really about integration .I would make the plea that we restore the adjective Integral, returning to the language of du Bois-Reymond and Hobson who recognized this as the Fundamental Theorem of Integral Calculus (FTIC), the theorem that tells us not that differentiation and integration are inverse processes to each other, but that integration has two very different manifestations. Recognizing and being able to take advantage of this equivalence is the true source of the power of calculus.
 Lacroix, S. F. 1802. Traité élémentaire de Calcul Différential et de Calculus Intégral. Paris : Chez Duprat
 Granville, W. A. 1904. Elements of the Differential and Integral Calculus. Boston : Ginn & Company
 du Bois-Reymond, P. 1876. Anhang über den fundamentalsatz der integralrechnung. Abhandlungen der Mathematisch-Physikalischen Classe der Königlich Bayerischen Akademia der Wissenschaften zu München. 12: 161–166.
 Hobson, E. W. 1907. The Theory of Functions of a Real Variable and the Theory of Fourier Series. Cambridge University Press.
 I do not see the need to define them as limit of Riemann sums with completely arbitrary tags (values at which the function is evaluated). Cauchy's left-hand sums or Granville's mid-point sums are perfectly adequate for the purposes of first-year calculus. Riemann introduced his sums with arbitrary tags to facilitate the study of totally discontinuous functions that are nevertheless integrable, a topic that is not appropriate for most first-year calculus courses.
 A very good progression of such projects based on computing moments (both mechanical and probabilistic) can be found in Smith and Moore's Calculus: Modeling & Application, 2nd edition, at www.math.duke.edu/education/calculustext/
 A nice example of such a problem appeared in the 2004 AP Calculus exam, asking for the position of a particle at time t = 2 if it travels along the x-axis at a velocity given by v(t) = 1 – arctan( exp(t) ) and it is at position –1 at time t = 0. Most students are stymied by the fact that they cannot find an explicit formula for the anti-derivative of this function, not recognizing that what they really need is the definite integral of the velocity over 0 <= t <= 2, and this can be determined to whatever accuracy is desired by a numerical approximation of the definite integral of 1 – arctan( exp(t) ).
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