, Volume 46, Issue 4, pp 691–695

The fundamental theorem of calculus: visually?

Original Article

DOI: 10.1007/s11858-014-0608-9

Cite this article as:
Kirsch, A. ZDM Mathematics Education (2014) 46: 691. doi:10.1007/s11858-014-0608-9


The paper wants to show how it is possible to develop based on an adequate basic idea (so-called “Grundvorstellung”) of the derivative a visual understanding of the (first) Fundamental theorem of Calculus.


Derivative Basic idea Fundamental theorem of calculus 

1 Introduction

“What about this is visual? I can’t see the reason for this supposedly astounding connection between the area and the derivative.” Breuker’s (1991) quote from a student (after their teacher showed them a “visual” proof) is a starting point for the following: our goal is to facilitate a deeper understanding of the relationship between the integral and the derivative through an appropriate perception of a derivative. This graphical understanding will then allow one to better comprehend the (first) Fundamental Theorem of Calculus:
$$ \frac{d}{dx}\mathop \int \limits_{a}^{x} f = f\left( x \right) $$
(for continuous functions f).

In doing so, the “visual” proof should not be seen as less thought-out or even as a stark contrast to a “formal” proof. Instead, the visual interpretation should, on its own, reveal the meaning of the theorem and lead to a deeper understanding of the concepts, essentially making, in this case, a formal proof—which, of course, has its own importance—appear subsidiary in the eyes of the students.

2 Foundations

From the beginning, we take a functional point of view by defining for a given function f in an interval, containing point x = a, the area collection function Fa (see Fig. 1).
Fig. 1

Fa shows the area “under” the curve

Fa(x) = area between a and x (x ≥ a), bounded by the graph of f and the x-axis such that Fa(x) is positive above the x-axis and negative below (calculated the other way around if x < a).

The naive concept of an area suffices as a foundation for the above definition of Fa, as, for example, Artin used in an introduction to the calculation of an integral (Kirsch 1976). For this reason, we are not naming Fa the “integral” function. But obviously Fa(x) is nothing else than an interpretation of the otherwise introduced integral
$$ \mathop \int \limits_{a}^{x} f = \mathop \int \limits_{a}^{x} f\left( t \right)dt. $$

We will sometimes call Fa(x) the “area under f” for short.

In any case, the assertion takes the form \( F_{a}^{'} \) = f. But what does \( F_{a}^{'} \) mean? This question is very difficult to answer if you only think of the derivative as the slope of the tangent line or, in other words, if you cannot view the derivative as the instantaneous velocity. The latter allows the interpretation of \( F_{a}^{'} \) as the rate of change of the area, which is much more natural here.

It doesn’t hurt to start by only looking at positive functions so that the representation of Fa as an area makes sense geometrically speaking, as this creates a more suggestive approach. One should, therefore, not have any qualms about introducing the key concepts in this way. The translation to functions with negative values will then barely pose any basic problems at all. In addition, this method provides good real-world examples such as that in which x is time, f(x) is the rate at which water flows into a container (empty at time a), and Fa(x) is the amount of water in the container at time x. Of course, even in this example, Fa(x) cannot be negative. In this regard, a more favorable interpretation is given by Bender (1990, p. 112).

3 The key insight

Next, we ask “How fast is the area growing if you travel down the x-axis at a constant rate? Imagine that the area under the curve is to be painted. How fast would you have to apply the paint? Where (on the graph of f) can you see this speed at a given point x?”

At this point, we can expect the students to come up with conjectures such as:
  • “You can see it on the right edge of the area, it depends on the height of the graph at point x.”

  • “The bigger the height, the faster the area changes.”

  • “The rate of change must be proportional to the height.”

The last of these conjectures already contains the central idea behind the validity of the Fundamental Theorem. “A physicist would even consider this a proof” (Kirsch 1976, p. 98).

Naturally there will be students who do not see the correctness of this last conjecture right away. In this case, one must be clear what is meant by the rate of change of the area: an estimate for this is the quotient of the change in area in a small time frame and this change in time. Because the x-axis is traveled along at a constant rate, you can “consider the time to be x.” (This means that the “velocity is one,” or 1 cm/s). Therefore:
$$ \begin{aligned} F_{a}^{'} ( x ) &\approx \frac{{{\rm area\,under} \; f \,{\rm between} \; x \, {\rm and}\; x + \Delta x}} {{\rm (small)\; time\,interval} \; \Delta x}\\ &\approx \frac{{{\rm area\,of\,rectangle} \; f( x )\cdot \Delta x}}{{\rm ( small )\; time\,interval} \; \Delta x} \\ &= f( x). \end{aligned} $$
After this, it is clear that
$$ F_{a}^{'} \left( x \right) \approx f\left( x \right) $$
And in the case f(x) = constant, even
$$ F_{a}^{'} \left( x \right) = f\left( x \right) $$
(for Δx > 0).

The formal precision of the outcome—with the definition “the limit as Δx → 0”—is at this point less important. We are striving first and foremost to build strong basic ideas of the concepts. It is enough, at first, to think of the time interval Δx as very small, as the physicists in such cases do: “so small, that by further divisions, the quotient doesn’t change.” Then the students will be convinced that \( F_{a}^{'} \) ≈ f(x) with a certain degree of confidence if Δx is made small enough, although this is no different from the statement \( F_{a}^{'} \) = f(x), omitting the limit.

What has been mentioned above leads without question to the desired conceptual insight into the validity of the Fundamental Theorem—as long as this “view” is not superficially restricted to geometric figures (see discussion at the beginning) and the “derivative” is not limited to the concept of the slope of the tangent line.

4 And the slope of the tangent line?

Now, one naturally would like to see what the Fundamental Theorem has to do with the slope of the tangent line: this with regard to this expanded concept of the derivative and the unclear wording of the Fundamental Theorem in the form “finding the slope of the tangent line and calculating the area are inverse operations.” According to Breuker (1991, p. 278), the most astounding and confusing part of the Fundamental Theorem for most students is this: “How are the concepts ‘tangent’ and ‘area’ related?”

The point is to interpret the rate of change of the area as a slope, not of the original function, but of the “area collection” function, that until now was not needed.

This requires the graphing of the function Fa, preferably on a separate coordinate plane. In Figs. 2 and 3, a = 0. F0(x) is denoted as the shaded area in Figs. 2 and 3 as the darkened line. (We will allow ourselves here and in the following to use the abbreviated phrasing that stems from the condition f > 0 and the assumption that 0 = a ≤ x. Also, the term “height” will refer to the function value f(x)).
Fig. 2

Graph of the function f(x)

Fig. 3

Graph of the corresponding “area collection” function F0(x)

The representation of an area by a height poses an often overlooked difficulty for students. It requires a translation skill that goes against natural geometric understanding: “square centimeters are going to be plotted as centimeters?” Equating areas and heights makes the assumption that, as is customary in mathematics, one ignores the units and views both simply as numbers.

Kaiser-Meßmer (1986, p. 98) shows that these difficulties occur even in upper-level mathematics courses, quoting students saying: “I don’t understand how an area can be equal to a length.”

Being comfortable with this type of translation is not only important for understanding the Fundamental Theorem, but it is also essential for further studies in Calculus. Suggestions for practice problems in this context (construction and recognition of the “area collection” function) can be found in Worksheets 1 and 2 at the end of this paper.

Students should always use simple area formulas to determine the “area collection” function as individual points first and then find a general formula.

Working with such problems eventually leads the students to the discovery of the astonishing relationship \( F_{a}^{'} \) = f. At first, this speculation will come from comparing the functions and then later with respect to “curve sketching” ideas comparing the graphs of Fa and f.

These observations alone are, of course, no proof, but rather only a verification of the already recognized theorem. In Breuker (1991) they are the starting point of a complete proof, where not only the large-scale behavior of the graphs of f and Fa are compared, but also the local relationships are broken down.

In contrast to this, I propose establishing this relationship to the slope of the tangent line by following systematically the correlation of the concepts with respect to f and Fa.

Fa(x) means:

  • on the one hand, the area under f up to position x;

  • on the other hand, the height of Fa at position x.

\( F_{a} \left( {x + \Delta x} \right) - F_{a} \left( x \right) \) means:
  • on the one hand, the area under f between x and x + Δx (Fig. 4);
    Fig. 4

    Area under the curve of f

  • on the other hand, the change in the height of Fa from x to x + Δx (Fig. 5).
    Fig. 5

    Change in the height of Fa

$$ \frac{{F_{a} \left( {x + \Delta x} \right) - F_{a} \left( x \right)}}{\Delta x} {\rm means:}$$
  • on the one hand, the average height of f in the interval [x, x + Δx];

  • on the other hand, the slope of the secant line from Fa in the interval [x, x + Δx];

Therefore, as Δx gets smaller and smaller, \( \frac{{F_{a} \left( {x + \Delta x} \right) - F_{a} \left( x \right)}}{\Delta x} \) means:
  • on the one hand, the height of f at position x;

  • on the other hand, the slope of the tangent line of Fa at position x.

In this way, one gains full insight more easily (and without a flexible imagination of “velocity of change”)—and one comes to a visual understanding of the Fundamental Theorem of Calculus, as demanded from the student quoted at the beginning. This is because also in this approach the intended “connection between the fundamental ideas of ‘tangent lines’ and ‘area’” (Breuker 1991, p. 278) is established.

In conclusion, the following is once more emphasized: before a theorem is proven, there have to be appropriate basic ideas of its meaning—an unreasonable demand for mathematics instruction, perhaps?

Copyright information

© FIZ Karlsruhe 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KasselKasselGermany

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