Infinitesimals-based registers for reasoning with definite integrals

Rob Ely University of Idaho

Abstract: Two representation registers are described that support student reasoning with definite integral notation: adding up pieces (AUP) and multiplicatively-based summation (MBS). These registers were developed in a Calculus I class that used an informal infinitesimals approach, through which differentials like dx directly represent infinitesimal quantities rather than serving as notational finesses or vestiges. Student reasoning reveals how the AUP register supports modeling with integral notation and how the MBS register supports sense-making with and evaluation of integrals. Keywords: calculus, integral, register, semiotics, infinitesimal, differential

My goal is to examine and illustrate two registers for interpreting and working with definite integral notation, registers that are particularly useful for supporting student modeling and sense-making with integrals. These registers—adding-up-pieces (AUP) and multiplicatively-based summation (MBS)—are situated in a Calculus I course that uses an “informal infinitesimal” approach to calculus. I briefly summarize this general approach to calculus before describing these registers and how students reason with them.

Informal Infinitesimals Approach to Calculus For nearly two centuries, Calculus was “the infinitesimal calculus.” For its inventors, G. W.

Leibniz and Isaac Newton, it was a set of techniques for systematically comparing infinitesimal quantities in order to determine relationships between the finite quantities that they comprised (and vice versa). In the 19th century, calculus was reformulated in terms of limits rather than infinitesimals, and 20th century calculus textbooks have followed suit. Yet calculus textbooks and courses still use Leibniz’ notation, dx and ∫ , but without the meanings Leibniz assigned to these: dx is an infinitesimal increment and the big S is a sum (“summa”). The notations are now vestiges, and in particular, differentials no longer directly represent quantities that students can manipulate and reason with.

The guiding principle of the informal infinitesimals approach is to restore this direct referential meaning to calculus notation. The approach is supported by the work in nonstandard analysis in the 1960s showing that calculus can be founded upon infinitesimals with equal rigor and power, but it uses the informality of Leibniz’ reasoning rather than the formal development of the hyperreal numbers (e.g. Keisler, 1986). For instance, the derivative at a point dy/dx really is a ratio of two infinitesimal quantities, not code language for lim!→!

! !!! !!(!)!

. The chain rule is canceling fractions. And an integral really is a sum of infinitesimal bits, each of which in MBS is given by the product f(x)·dx. By taking differentials seriously, students can develop formulas for volumes of rotation, arclength, work, and many other ideas, formulas, and applications in first-year calculus (Dray & Manogue, 2010), and in vector calculus and physics (Dray & Manogue, 2003).

Registers and Signs For Duval (2006), a representation register is a collection of signs and a set of

transformations by which some of these signs can be substituted for others. Transformations within the same register are treatments; transformations of signs from one register to another are conversions. Barthes defines a sign as a combination of a signifier and a signified (1957/1972). For instance, a bunch of roses (signifier), together with the concept of passion it is representing (signified), comprise a sign. In our case, when a mathematical representation (e.g., “dx”) signifies a concept (e.g., an infinitesimal increment), the combination of the representation “dx” (signifier) and infinitesimal increment (signified) is a sign. An interpretation is thus a signified concept. So if a representation stays the same but its interpretation changes, it becomes a different sign, since it signifies a different thing or concept. This points to a conversion to a different register, because within a given register interpretation should remain relatively stable. Such a conversion is often accompanied by a new lexicon of signs, interpretations, and treatments that which might support the new purpose or apply to the new context.

Consider the following example: We can use infinitesimals to develop the formula for the arclength of a curve in the plane between x = 0 and 1. We imagine a curve to be comprised of infinitesimal segments, each of which is the hypotenuse of a right triangle with legs dx and dy. Then the arclength of the curve would be the sum of these hypotenuse lengths:

𝑑𝑥! + 𝑑𝑦!!!!! . So far we have performed a modeling step, treating the dx and dy as quantities

representing magnitudes, and have used the adding up pieces (AUP) register (which I detail soon). But this is no form to be evaluated for any particular curve, however. To be evaluated, the integral must first be converted by imagining all the dx’s as uniform in size and then being

factored from the integrand, to get 1+ (!"!")!  𝑑𝑥!

!!! (if y is a function of x, say g(x)). Now the

integral is of the form 𝑓(𝑥)  𝑑𝑥!!!! (where f(x) is 1+ 𝑔′(𝑥)!). So the integral can be evaluated

by F(1) – F(0), for some F as an antiderivative of f. Thus using algebraic manipulations we have converted to a new register that is suited for evaluating integrals. The [  ]! + [  ]! structure lost significance and the 𝑓(𝑥)  𝑑𝑥!

! structure gained salience, and it became important for the original curve to be seen with y as a function of x.

The Adding Up Pieces (AUP) and Multiplicatively-Based Summation (MBS) Registers

The elements of the AUP and MBS registers are based on the work of Jones (2013, 2015a,

2015b). The interpretations involved in both registers are summarized in Figures 1 and 2, and the strange numbering on these (I2, etc.) draws from the learning progression in my class through which they emerge, which I reference elsewhere in detail (Ely, in review). The interpretations in the AUP register (Figure 1) entail the idea that a definite integral measures how much of some quantity A is accumulated over an interval of a domain, say from t = a to b. This domain is partitioned into infinitely many infinitesimal increments of uniform size dt. For each infinitesimal increment dt there corresponds an infinitesimal increment of A, dA. The integral  !! adds all of these up to give the total accumulation of A over the interval from t = a to b. In

order for this to make sense, one must appeal to conception C5: The sum of infinitely many infinitesimal bits is a finite accumulation of A.

The AUP register allows one to transparently represent a general bit of the sought quantity and of the whole quantity as an accumulation of these bits. The treatments in the register include (a) writing a symbolic expression for a generic bit dA and (b) rewriting this expression for dA in terms of other infinitesimal quantities that specify the expression for the domain at hand, usually in terms of its corresponding domain increment dt. These treatments require viewing infinitesimals as legitimate quantities that behave normally under algebraic operations (including possibly some additional Leibnizian rules for operating with infinitesimal quantities). The treatments also rely on two other grounding conceptions: (1) equivalent expressions can be substituted for the same quantity (conception C6), and (2) a foundational understanding of covariation, which in turn relies on the basic understanding that variables vary (Thompson & Carlson, in press). The student must be able to coordinate changes of A with changes of t in order to reason that for each increment dt there is a corresponding increment dA.

The MBS register includes many of the same notational interpretations as AUP, but it also adds to these the expression of each piece dA as a product r(t)·dt. This introduces the integrand, which is necessary for the Fundamental Theorem of Calculus (FTC) to apply. We follow Thompson, Byerley, & Hatfield’s (2013) approach to treat the integrand r(t) as a rate at which A accumulates over the increment dt. This ultimately allows us to recognize and use that an accumulation function f for A will have r(t) as its rate-of-change function. Thus, one supporting conception in this interpretation of the product r(t)·dt is the idea that A accumulates at a constant rate r(t) over the dt increment, and that this constant rate is determined by the value of t closest to that dt increment. This relies on conception C4: A concept image for rate of change at a moment (Thompson, Ashbrook, & Musgrave, 2015). This momentary rate of change can vary

constantly as t varies. The second supporting conception is the multiplicative structure associated with rate, expressed in conception C1i (see Figure 2). The

Figure 1 – Interpretive elements of integral notation in the adding up pieces (AUP) register

Figure 2 –Interpretive elements of integral notation in the multiplicatively-based summation (MBS) register

r(t)·dt

I1i. dA: a generic “little bit” of the quantity A

I2i. An infinitesimal increment of t

I3i. “Rate” at which A accumulates over the dt-sized increment of t

I4i. Sum of all the bits of A, as t hops along by dt-sized increments from a to b

C1i. The multiplicative structure: If A accumulates at a rate of r(t) A-units per t-unit, over an increment of dt t-units, the product r(t)·dt is the accumulated bit of A.

a

b∫

C5i. The sum of infinitely many infinitesimal bits is a finite accumulation.

word “rate” is used here broadly. It does not necessarily mean that r(t) is measured in a compound unit like miles-per-hour. Rather it means that as t changes, A changes by a proportional amount, and r(t) provides that rate of proportionality (Lobato & Ellis, 2010). Most broadly, r(t) serves as a factor allowing conversion from an increment of t to an increment of A, by dA = r(t)·dt. Reasoning with quantities, rather than bare numbers or symbols, is crucial to this interpretation.

The MBS register promotes reasoning with the FTC: if we can find any accumulation function f(t) whose rate-of-change function (i.e. derivative) is this r(t), we can use it to recover the accumulated amount A, by determining f(b) - f(a). The treatments within the MBS register include the same kinds of algebraic operations with summand and integrand as in the AUP register, and also include evaluating the integral f(b) - f(a) by means of an antiderivative f.

Other Modes of Student Reasoning with Definite Integrals

Although we focus on AUP and MBS, these modes of reasoning are relatively rare among

students in traditional calculus courses. For instance, Jones (2016) surveyed 150 undergraduate students who had completed first-semester calculus, using Prompts 1 and 2 on the next page. Only 22% of students made even a passing reference to summation of any kind on either prompt, and on each prompt less than 7% appealed to reasoning consistent with AUP or MBS. On the other hand, 87.3% of students appealed to an “area” interpretation on Prompt 1, and 76% used an “anti-derivative” interpretation on Prompt 2.

These two interpretations have also been described, and found prevalent, by other researchers. The area interpretation is that the definite integral represents an area “under” a curve in the coordinate plane, with the “d[]” denoting the variable on the horizontal axis, which forms the bottom of the shape. “The shape is taken as a fixed, undivided whole that is not partitioned into smaller pieces” (Jones, 2015b, p. 156). The anti-derivative interpretation is that the integrand came from some other other “original function” through differentiation; now the integral symbol represents an instruction to find this original function. The d[] dictates the independent variable “with respect to” which the derivative had been taken, and the limits of integration are the values that one must plug into the original function to get the numerical answer (Jones, 2015). Fisher et al. (2016) found that the majority of students in a standard calculus class used only the area interpretation when describing the meaning of a definite integral, and Grundmeier, Hansen, & Sousa (2006) found that only 10% of students mentioned an infinite sum when asked to define a definite integral.

Various studies claim that sum-based interpretations of the definite integral are much more productive in general for supporting student reasoning than are area and anti-derivative interpretations (e.g., Sealey, 2006, 2014; Sealey & Oehrtman, 2005, 2007; Thompson & Silverman, 2008, Jones 2013, 2015a, 2015b; Jones & Dorko 2015; Wagner 2016). For modeling in particular, the area and anti-derivative interpretations have serious limitations. The area interpretation is problematic when modeling in the myriad situations when the sought quantity is difficult to imagine as the area of a region (e.g., work, velocity, force, volume, arclength) (Thompson et al, 2013; Jones 2015a). The anti-derivative interpretation provides even less support for modeling, since it gives only a technique for evaluating a definite integral, not for creating one (Jones 2015a). These interpretations produce significant obstacles for students modeling with integrals in physics applications (e.g., Nguyen & Rebello, 2011).

Along with AUP and MBS, there are other sum-based interpretations of integral notation, notably the Riemann sum (limit of sums). Nearly all calculus books define the definite integral using Riemann sums, but this fact seems to contribute little to building sum-based reasoning for the students who use these books. When investigating this apparent pedagogical disconnect, Jones, Lim, and Chandler (2016) found that instructors’ teaching moves lead students to perceive the limit of Riemann sums not as a conceptual basis for understanding the definite integral, but merely as a calculational procedure that allows an integral to be estimated accurately. Another way that the limit process involved in the Riemann sum interpretation can form a conceptual obstacle for students is through the problematic collapse metaphor, through which students imagine the pieces losing a dimension in the limit, so the d[] loses its quantitative meaning (Oehrtman, 2009).

Data Collection

I taught an experimental Calculus I class using the informal infinitesimals approach, for

science, engineering, and math majors at a large public university in the northwestern U.S. I conducted semi-structured interviews and analyzed student written work. I focus here on two prompts I used in the interviews: Prompt 1, which is verbatim from Jones (2013, 2016), and Prompt 3, a novel modeling context of a kind very different from what the students seen before (although they had done a few volume-of-rotation problems in class and on homework).

Prompt 1: Explain in detail what 𝑓 𝑥 𝑑𝑥!!

means. If you think of more than one way to describe it, please describe it in multiple ways. Please use words, or draw pictures, or write formulas, or anything else you want to explain what it means.

Prompt 3: Set up an integral that represents the volume of this solid, whose base is the region bounded by the curves y=√x and y=-√x, and whose cross sections perpendicular to the base and perpendicular to the x-axis are squares.

Results

I analyze here the reasoning displayed in the responses of two students, Dmitri and Galena.

Reasoning in the AUP Register In response to Prompt 3, Dmitri’s initial answer of ( (2𝑦 ⋅ 𝑑𝑥)!!

!!! ) was incorrect, but after reflecting for a minute he corrected it to (2𝑦)! ⋅ 𝑑𝑥!

!!! . He notes that if the slice was (2ydx)3, it would make a perfect cube, which can’t be right. It should instead look like what is in Figure 4. He narrates as he draws and labels the slice: “This [indicates the width] is going to be dx. This will be the same thing as the other one: this one [indicates the slice’s height dimension] is still 2y. This one’s still 2y [indicates the slice’s depth dimension]. But the width is still dx. So to find the volume of that, we’d have 2y squared times dx. And that is all. And that solves my problem.” He then describes how the slices are aggregated, each time “you’d go up an infinitely small

430 Chapter 6 Applications of Integration

11. The solid with a semicircular base of radius 5 whose cross sec-tions perpendicular to the base and parallel to the diameter are squares

12. The solid whose base is the region bounded by y = x2 and the line y = 1, and whose cross sections perpendicular to the base and parallel to the x-axis are squares

squarecross section

base

y ! x2

y

x

13. The solid whose base is the triangle with vertices 10, 02, 12, 02, and 10, 22, and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles

14. The pyramid with a square base 4 m on a side and a height of 2 m (Use calculus.)

15. The tetrahedron (pyramid with four triangular faces), all of whose edges have length 4

16. A circular cylinder of radius r and height h whose axis is at an angle of p>4 to the base

hr

circularbase

d

17–26. Disk method Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when R is revolved about the x-axis.

17. y = 2x, y = 0, x = 3 (Verify that your answer agrees with the volume formula for a cone.)

x

y

3

y ! 2x

R

(3, 6)

0

5. Why is the disk method a special case of the general slicing method?

6. The region R bounded by the graph of y = f 1x2 Ú 0 and the x-axis on 3a, b4 is revolved about the line y = -2 to form a solid of revolution whose cross sections are washers. What are the inner and outer radii of the washer at a point x in 3a, b4?

Basic Skills7–16. General slicing method Use the general slicing method to find the volume of the following solids.

7. The solid whose base is the region bounded by the curves y = x2 and y = 2 - x2, and whose cross sections through the solid per-pendicular to the x-axis are squares

x y

8. The solid whose base is the region bounded by the semicircle y = 21 - x2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares

yx !1 ! x2 y "

yx

9. The solid whose base is the region bounded by the curve y = 1cos x and the x-axis on 3-p>2, p>24, and whose cross sections through the solid perpendicular to the x-axis are isosceles right triangles with a horizontal leg in the xy-plane and a vertical leg above the x-axis

y

xy ! cos x

y

x

10. The solid with a circular base of radius 5 whose cross sections perpen-dicular to the base and parallel to the x-axis are equilateral triangles

yx

equilateral triangles

circular base

y ! 2 " x2

y ! x2

x y

M06_BRIG7345_02_SE_C06.3.indd 430 21/10/13 5:20 PM

!

x

y

amount and then you’d do the same thing for that one, and you’d do that for all numbers between 0 and 1.” Then he notes that the collection of all these pieces is the volume of the whole figure.

In this sequence of reasoning, Dmitri appeals to all the elements of AUP. He has imagined a domain partitioned into increments of infinitesimal size dx (I2), described a representative slice of the figure’s total volume as the thing being summed (I1), and described the integral as the sum of all such pieces across the appropriate domain (I4). He notes that these have infinitesimal volume but that when you sum them all you get the whole region, which indicates he is using C5.

Dmitri checks his answer by appealing to dimensional quantities and units: even if dx is a “really small amount of meters, it’s still meters – so meters squared times meters equals meters cubed, and that’s the unit of volume.” Dmitri uses AUP, not MBS; he never seems to need the summand to be in the form “f(x)·dx.” His initial answer is not at all in that form, and his final answer still does not have the integrand written as a function of x. Additionally, he appeals to multiplicative structure when he talks about the summand, he does not describe or treat the area part, the integrand, as a “rate” at which the figure’s volume grows with each infinitesimal increment of the domain.

Reasoning in the MBS Register In response to Prompt 1, to explain what 𝑓 𝑥 𝑑𝑥!

! means, both Galena and Dmitri express notational interpretations I1-I4 and conceptions C1i and C5i. Galena’s succinct response is shown here, and I indicate how it displays I1-I4. There is no textbook with MBS in it yet, but if there was, Galena’s account would be the “textbook” description of MBS:

!

!

!!

!

I2i"!

I3i"!

I1i"!I4i"!

G: Okay, so, I’ll just separate this into chunks: So the dx is gonna be a small increment in time, like ideally it would be infinitely small. Um, so this [gestures to the “dx”] is a chunk of time, or whatever is on your x-axis. It doesn’t necessarily have to be time; it could be meters if you were doing it in length. But it’s a small increment of whatever x is. Then f(x) is the rate at which that grows over this [points to the dx] chunk of time, per se. And then, so this is a rate [points to the f(x)]. So it would be like meters per second, or whatever this x value is per whatever is the y value [she says these reversed but writes them correctly]. Uh, and this [points to the entire integral] is making this a summation of these chunks. So this [points to f(x)dx] is going to be a chunk. And this is the summation from a to b of those tiny chunks that you’re adding up along the way.

Figure 4 -- Dmitri’s modeling of a representative piece of volume. The axes labels were written by the interviewer.

Dmitri’s responses to Prompt 1 also illustrate his reasoning using I1-I4, C1 and C5, and he explicitly refers to f(x) as a “rate of change function; … let’s say it’s meters per second, then dx could be a really small increment, an infinitesimally small increment, of seconds.”

Converting From the AUP Register to the MBS Register An example of Galena’s written work illustrates the process of converting between the two

registers. The problem asks her to first set up, then evaluate, an integral representing the volume of the figure created by rotating around the x-axis the region enclosed by the curves y = x + 6 and y = x2. Galena makes a couple of small mistakes in her work, but the switch between registers is clear. Her modeling work, to set up the integral, is shown in Figure 5.

To evaluate the integral, Galena converts to interpreting the integrand [𝜋(𝑥 + 6)! − 𝜋(𝑥!)!]

not as two dimensions of a slice but as a “rate of change,” (she writes this). She then seeks to evaluate by finding an “accumulation function” evaluated at the starting and ending values of 0 and 3. The interpretation of the integrand has changed with the register shift, so she no longer refers to elements of the figure.

Discussion

These are a few illustrative examples of how students in an informal infinitesimals calculus

course used the AUP register to model with definite integrals and the MBS register to reason with and evaluate definite integrals, and how there is an explicit change of interpretation marking the conversion between the registers. Since the two registers support these distinct purposes, it may support student learning for the instructor to teach the registers independently and to be explicit about the interpretations and affordances in each register. By explicitly teaching students the signs (notations and interpretations), treatments, and purposes of the two registers, we may help them develop meta-level awareness of the significance and affordance of their actions in the registers. An informal infinitesimals approach to calculus can help students develop these two registers, which are more powerful tools for reasoning with definite integral notation than the prevalent antiderivative and area interpretations.

Figure 5—Galena’s modeling in the AUP register

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