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When one wants to let students calculate (Riemann) integrals in calculus, what is a good term to call this task?

If you want to focus on what the main task is, you may call it "Calculate the definite integral" or "Integrate a function", but as long as the function is continuous, one might use terms like calculating the "primitive function" or "antiderivative" (In Germany the only science-related term is "Stammfunktion" (=primitive function), but some people came up with the non-scientific term artificial term "Aufleitung" (maybe, a good "translation" would be "ascrivative") meant to be the opposite of a derivative ("Ableitung" in German)). However this covers only a part of real world integrals, since the concept of integration is much more general than only being the opposite of taking derivatives.

What are (dis)advantages of using terms like "antiderivative" instead of "integrating" or "primitive function"?

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    $\begingroup$ I never heard "Aufleitung" so far. What a strange term. The "auf/ab" dichotomy makes no sense at all in that context. (I understand you do not endorse it, but I just needed to complain about this word.) $\endgroup$ – quid Mar 27 '14 at 13:18
  • $\begingroup$ I came across "Hochleitung" as well. I don't like any of these terms. Strangely enough, I'm OK with antiderivative. $\endgroup$ – Roland Mar 27 '14 at 14:45
  • $\begingroup$ @Roland I think with "anti" it is atleast a clear negation. My main issue is I think that neither "hoch" nor "auf" feels like a reasonable negation of or opposite term to "ab" in this context. If this catches on to add might soon be called "aufziehen" or "hochziehen" :-) [For some readers, I likley should add that to subtract is sometimes called abziehen.] $\endgroup$ – quid Mar 27 '14 at 16:36
  • $\begingroup$ I agree with quid. I never heard "Aufleitung" or "Hochleitung" to mean integration, and I hope not to hear them in the future. $\endgroup$ – Andreas Blass Mar 30 '14 at 3:12
  • $\begingroup$ Fortunately (for me and students), I don't have to lecture in German, but if I did, I'd probably use "integrieren" (and "differenzieren"). Are those wrong? Would they immediately mark me as an American butchering the German language? $\endgroup$ – Andreas Blass Mar 30 '14 at 3:15
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Foreword

Students in secondary education will not leave the space of Riemann integrals (and symbolic Integrals). The existence of different integration concepts in higher maths is totally irrelevant here.

Knowledge

The operations calculate the derivative and calculate the integral (whether Riemann indefinite or symbolic) are inverse operations with some restrictions (one-way is translation invariant and thus not injective, discontinuities). These are to be made clear to the students.

Disregarding historic context, these restrictions are the only disadvantages of inverse name pairs for the two operations.

If you regard the historic context that created non-inverse names or not, is up to you. If you do regard the historic context, it's highly disadvantegous not to use the actual, history-driven names.

Communication competence

Students need to be capable to read texts and understand people which use the actual names.

If students use inverse name pairs, one of them should be the actual name, so that the people they talk to can understand them. People not accustomed to this problem will probably be confused at first but able to understand later-on.

The disadvantage of using inverse name pairs is thus only serious, if the students lack passive understanding of the actual names.

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