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David Bressoud has a 2019 book that argues for teaching integration before derivatives, and only later teaching limits, partly because that matches the way that the ideas developed historically: "Calculus Reordered: A History of the Big Ideas".


Because: And now I'm padding my answer with 21 characters.


As with a huge number of the "why do we do this order" questions, or the "why don't we teach real analysis before calculus" questions, they seem to tacitly assume that the order of teaching is based on philosophical explanation or even formal mathematical proof. WRONG! This is NOT the reason why instruction is structured the way it is. ...


I'd like to comment on why, indeed, it would be reasonable to present the subject with integrals first... (which is not what is done, nearly universally, I understand). In particular, I'd argue that the notion of "derivative" is more sophisticated than "area under a curve". This includes the subtlety of "instantaneous rate of change&...


One issue is that for differentiation, you can find the derivatives of so-called "elementary functions" (e.g. powers, exponentials, logarithms, trig functions) directly from the definition of a limit, and there are then standard ways to find the derivatives of an expression knowing the derivatives of its component parts (e.g. the product and ...


I think one of the main reasons for the ordering, limits-> derivatives -> integrals is that it follows in order of difficulty in some sense of the word. In reality the situation is more complex, but if you start with the minimal (read $\epsilon$-$\delta$) definition of a limit it's fairly straightforward. It's also almost essential for even a semi ...


The text I know of that does integration first is: Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. An advantage of using this for top-rate incoming university students is that they do not think "I already know this" and tune out [as they may do when the course begins with differentiation].


While there are no theoretical difficulties with developing integration first ("from scratch" measure theory books demonstrate this), it presents some pedagogical challenges. Most problems in existing Calculus courses have "closed form" solutions. It is rare to just set up a problem and then numerically evaluate. The primary tool ...

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