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Does anyone have any good reasons to offer for introducing definite integrals before indefinite integrals, or the other way around? Definite integrals first because they have a clearer geometric interpretation? Indefinite integrals first because they can be defined without introducing the Riemann sum? Definite integrals first because they make it easier to understand the logic behind the Leibniz notation?

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It seems perfectly clear to me that the rather simpler-to-calculate indefinite integral be introduced before the double-substitution definite integral. I typically introduce integration as solving a differential equation, then introduce notation later and the fundamental theorem of calculus as a surprising and amazing fact afterwards. –  AndrewC May 26 at 20:34

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In my opinion what is important is that one defer use of the term indefinite integral until after one has seen definite integrals and the Fundamental Theorem of Calculus. If you hear about "indefinite integrals" first, then it is only natural to cognitively overlay definite integrals on top of indefinite integrals: it's an indefinite integral plus only a little more: $F(x)(+C) \mapsto F(b) - F(a)$. In other words, this seems to abet what is perhaps the single most frustrating cognitive process of calculus students: the tendency to regard the Fundamental Theorem of Calculus as a definition.

To help with this I try to use the term antiderivative instead of "indefinite integral" as much as possible. At some point I slip up because after all we are using the same integral sign for both: I almost wish we didn't. (Using almost identical notation for two a priori incredibly different things which in the setting of the FTC become almost the same is one of the brilliant notational innovations of calculus. It's up there with the in/famous $\frac{dy}{dx}$.) I also begin discussion of antiderivatives while doing differential calculus. Whenever you define an interesting operation in mathematics, it is an easy sell to consider the corresponding inverse problem: what inputs, if any, lead to a given output? For me, the uniqueness of antiderivatives up to a constant is one of the major applications of the Mean Value Theorem. Generally at that time I mention the problem of existence of antiderivatives of any continuous function and allude to the fact that to solve this problem we need integration, by which I mean area functions.

(For many years I thought that you really did "need" integration to prove the existence of antiderivatives of continuous functions. Rather recently I learned that there are proofs of this which do not use integration whatsoever. The first person to give such a proof was apparently...Henri Lebesgue! As usual there is a nice MONTHLY article about this...)

I also start with the notation $\int_a^b f$ for "definite integrals". I tack on the $dx$ only after doing FTC and introducing integration by change of variables (or, as I was taught in high school and still think of it in the one variable context: "u-substitution"), at which point I try to explain the sense in which the $dx$ is "just" a brilliant Leibnizian device to streamline the change of variables process. (May the proponents of nonstandard analysis forgive me: I do not say anything about infinitesimals in my freshman calculus courses.)

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+1, but re $\int_a^b f$ ... ugh. This notation has the wrong units, e.g., if $f$ is a velocity in m/s, then we need the notation to be $\int_a^b f(t) dt$ so that the seconds cancel out and give us meters. –  Ben Crowell May 20 at 1:32
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"perhaps the single most frustrating cognitive processes of calculus students: the tendency to regard the Fundamental Theorem of Calculus as a definition." This is so true. In some sense, they are very abstract in their thinking. –  James S. Cook May 20 at 2:22
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@BenCrowell You're regarding the $\int$-symbol as sum kind of summation. Summing doesn't change the unit. But the definite integral is not a sum of evalutated integrands. The notation without differential is common for symbolic integration. –  Toscho May 20 at 4:57
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Maybe Toscho or Pete Clark would like to start a separate question on math.SE on this topic. I'm convinced I'm right, but I don't think this is going to be clarified in a comment thread. In addition to having the wrong units, $\int_a^b f$ breaks the grammatical rules of the Leibniz notation. Students need to learn this grammar, and they won't learn it if they don't see it modeled correctly. –  Ben Crowell May 20 at 15:09
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In fact, upon reflection, the notation $\int_a^b f$ is used in the celebrated calculus text of Spivak, which I have used for teaching an (honors) calculus class. Michael Spivak has about as much acumen and experience in calculus -- its theory, its pedagogical aspects, its connections to other parts of mathematics and also to physics -- as anyone I have ever known. So that's a more than sufficient defense of the notation from my perspective. –  Pete L. Clark May 20 at 16:21

A remark that has little to do with my answer: I believe Apostol's calculus does definite integration before differentiation.

The application of indefinite integration to initial value problems is interesting. For example, generically, we can ask: what position function $x(t)$ gives rise to a particular velocity function $v(t)$. So, we're looking for a function $x(t)$ which when differentiated gives us $v(t)$. This is just antidifferentiation. Although, I'm fairly sure it's wrong to say given $v(t)$ find $V(t)$. I think such problems are interesting and accessible before any discussion of the Riemann integral has been given. If you choose to also discuss the indefinite integral here the very nice thing you get to say: "this $c$ we add is to account for all the possible initial conditions." This seeds intuition for later work on differential equations in my view.

From a course management perspective, these give me a chance to get the students to study antiderivatives before we cover the Fundamental Theorem of Calculus (FTC). This means, as I'm teaching FTC there is a greater chance they understand what I mean when I say "antiderivative". I suspect they are more confused about $\sum$ notation and limits of sequences by the time I get to FTC. It's ironic that we worry about these minor points and yet feel free to drop sequential limits on them when so many have not seen a sequence in years. Backgrounds vary I suppose.

Something I've worked out for the sake of showing them what not to do is an indefinite integral from an explicit calculation of the Riemann integral with varying bound. It's not too much trouble to show: $$ \int_{0}^x dt = x \qquad \int_{0}^{x} t \, dt = \frac{1}{2}x^2 $$ from the definition. You can derive these from the known formulas for sums. Then, I ask them, can you work out $\int_{0}^{x} \sin (t) \, dt$? There is a class of students who needs to realize why antidifferentiation is necessarily educated guessing. This discussion helps with that. Annoyingly, I can't seen to find the explicit examples in my posted notes, if you're interested let me know, I'll find them. For more on how I present antiderivatives etc... you might look at: My Oldschool Calculus Notes circa 2010.

( I am happy to email you LaTeX source if you want it, however, it has some institution specific comments and my horrible grammar, I'm working on an edit this summer if my student comes through as I hope...)

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"I believe Apostol's calculus does definite integration before differentiation." That's an interesting remark. It's not so crazy: in graduate level real analysis one certainly does integration before differentiation, and one gains an appreciation that the latter really is a more difficult and delicate process than the former. –  Pete L. Clark May 20 at 0:26
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"Antidifferentiation/Integration is educated guessing." This is one of the most important meta realizations. Students come from school thinking math is straight forward, deterministic calculation. No, it's solving problems, which is often easiest by guessing a solution and proving it right. –  Toscho May 20 at 5:04
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Sometimes, students have "integrated/antiderived" before the math teacher introduces it: In physics, you need to solve differential equations if covering capacitors, inductors or harmonic oszillations. –  Toscho May 20 at 5:07
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To be fair, there are deterministic algorithms for integration of elementary functions (Risch Algorithm, among other extensions of it), but it's somewhat intractable for humans to do regularly. –  Jsor May 21 at 5:10
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@JamesS.Cook if you allow complex numbers (and provide a proper mapping of $sin(x)$ to its definition) it CAN integrate $sin(x)dx$, but I'm pretty sure 99% of the implementations of it have it programmed as a canonical example for performance if nothing else. –  Jsor May 21 at 5:37

Both history and applications suggest doing definite integrals first.

The applications of calculus to areas, volumes, moments, probabilities and expectations are all primarily definite integrals.

The history of things which look like calculus begins with Archimedes's calculations of areas, which are again definite integrals. And the things that developed first are usually the most obvious and most natural starting points.

If you do enough definite integrals numerically, it's easy to appreciate the utility of indefinite integrals afterwards.

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While I agree on the history, it is often that the most direct path to understanding/proof is wildly different from the historical development. I've never been formally introduced to Archimedes' arguments, for instance. –  vonbrand May 24 at 11:18
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@vonbrand, I agree: the oldest proofs are often not the easiest to understand. By "things", perhaps I should have said "concepts and definitions and theorems and applications." Meanwhile, you might enjoy some Archimedes! E.g.: cs.xu.edu/math/math147/02f/archimedes/archpartext.html –  Matt F. May 24 at 12:27

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