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Is there a canonical "What's the intuition behind integration?" question? If not should we declare one? Either case here and here are two example of this question on SE. There are probably more.

Update — Those questions have been merged, and I posted an answer.

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    $\begingroup$ Is this question—Why is the area under the curve the integral?—also relevant? $\endgroup$
    – Joe
    Jun 29, 2021 at 18:38
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    $\begingroup$ @Joe maaaybe, but I'd say the intuition behind integration is more general than the geometric interpretation. $\endgroup$
    – Mike Pierce
    Jun 29, 2021 at 18:45
  • $\begingroup$ This maybe? If not, what else are you looking for apart from geometric interpretation? $\endgroup$
    – Alexander Gruber Mod
    Jun 29, 2021 at 19:17
  • $\begingroup$ @AlexanderGruber maybe that (Is asking for intuition on integration the same as asking for intuition on the FTC?) I'd say intuition on integration should be about adding up, accumulating something, accumulating the thing that the area under the curve represents—for best intuition that something should thought of as more than just area. $\endgroup$
    – Mike Pierce
    Jun 29, 2021 at 19:36
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    $\begingroup$ This is a good question! I am currently designing a sequence of short videos on this theme for my coming vector calculus course. My observation has been that students (or their calculators, or WA...) can calculate integrals well enough, but the goal is surely that they should understand what this integral actually calculates! I want to drive home the point of splitting the "thing you want to calculate" into small parts and add them up to get an estimate. Then make the parts smaller to get a better estimate. You know the drill. $\endgroup$
    – Jyrki Lahtonen
    Jun 30, 2021 at 6:08
  • $\begingroup$ (cont'd) Yes, Riemann sums are the abstraction of this, but the students don't want to concentrate on that. They just want the tool, and miss out on a few key things. The challange I face is to make them see the key ingredients. Yes, the integral concept is easier to prove rigorously using upper/lower sums, but it comes with a price. $\endgroup$
    – Jyrki Lahtonen
    Jun 30, 2021 at 6:10
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    $\begingroup$ Hmm. This theme might also make a good question at MathEducators.SE. $\endgroup$
    – Jyrki Lahtonen
    Jun 30, 2021 at 6:11
  • $\begingroup$ Anyway, it may be difficult for the Math.SE format to naturally develop a "canonical" Q&A that serves all the needs about the intuition. If you volunteer to develop one, go ahead :-) $\endgroup$
    – Jyrki Lahtonen
    Jun 30, 2021 at 6:16
  • $\begingroup$ A bit more about my plan. Item 1: Area under a curve as an integral of the height, Item 2: Volume as an integral of the cross section area. Item 3: Volume under a surface as an iterated integral of the height. Item 4: Mass inside a volume as an integral of the density (3-fold iteration). Item 5: Population in an area as an integral of the density.... Anyway, the general process and the relation between the integrand and what the integral calculates is what I want to teach. $\endgroup$
    – Jyrki Lahtonen
    Jun 30, 2021 at 6:22
  • $\begingroup$ @JyrkiLahtonen Instead of developing a new canonical question—since no one pointed out a post I wasn't aware of—I'd take this one as the canonical question (it's got the best answer and most views), do a bit of editing/cleaning/linking, and request the other post be merged. $\endgroup$
    – Mike Pierce
    Jun 30, 2021 at 16:11
  • $\begingroup$ @JyrkiLahtonen I agree with your idea of what sort of intuition we should be relaying to students, and your latest comment has some great examples. :) Let me know if want to type up your thoughts in an answer on that post, otherwise I will type something up (sometime) and will draw on what you're talking about. $\endgroup$
    – Mike Pierce
    Jun 30, 2021 at 16:13
  • $\begingroup$ The question might currently deter people not into electromagnetic theory/physics. Integration as taking the expected value in probability theory would be a weird answer to this question but if this is supposed to become a general "intuition for integrals" questions one answer should probably take that angle $\endgroup$
    – Felix B.
    Jul 2, 2021 at 20:18
  • $\begingroup$ @FelixB. Not anymore ;) (that detail was irrelevant to the base question after all) $\endgroup$
    – Mike Pierce
    Jul 4, 2021 at 0:30
  • $\begingroup$ @MikePierce well then ... math.stackexchange.com/a/4190126/445105 :) $\endgroup$
    – Felix B.
    Jul 4, 2021 at 15:47

2 Answers 2

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Since nobody has pointed out an existing one, I hereby declare the question

What's an intuitive explanation for integration?

to be the canonical such question, and ideal target for duplicate flagging, henceforth.

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I don't know about a canonical one, but here's one I wrote a few years ago, perhaps more suitable for an analysis student than a beginning calculus student. But the language is extremely plain and, I think, clear:

Understanding the definition of the Riemann Integral

It does at least have the pictures showing a lower sum and an upper sum and why they will (normally) tend to converge as we refine the partition. In case someone else wants to use them, I hereby release the diagrams into the public domain.

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    $\begingroup$ But that explicitly asks for the definition and not the intuition. In some sense it is complement to the other question. The other one tries to answer why you would even want to know the area under a curve, this one answer how you would know the area under a curve $\endgroup$
    – Felix B.
    Jul 12, 2021 at 7:31

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