integration - calculus essay assistance

I am writing an essay for my calculus class and one of the requirements to meet within the essay is to demonstrate an understanding of integration by explaining a metaphor that uses integration.

This is the passage that I think meets that requirement but I am not sure if I should expand more on integration just to be sure:

To a person familiar with integration attempting to relate the metaphor back to math, this statement likely brings to mind images of their first calculus instructor drawing rectangles below a function when showing the class how to calculate the area under a curve. The reason Tolstoy’s statement conjures this reminiscent math memory to is because the two concepts being discussed are abstractly identical. Just as the wills of man that direct the compass of history are innumerable, so are the number of rectangles that are required to be summed to get an exact measurement of area under a curve. Despite the impossibility of calculating an infinite amount of something we must still calculate some amount of it if we wish to obtain the valuable information an approximation can provide.

For reference, here is the metaphor I am writing about:

"The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history. . . . Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history"

Could anyone provide some feedback? thanks!

Answer

In my opinion, if this is a serious assignment, then it would be a very difficult one for most students. In order to write something really solid, one needs to (i) have strong general essay-writing skills (this is an unusually difficult topic), (ii) have a very solid theoretical grasp of calculus in order to be able to compare metaphors with theorems and (iii) be able to merge the humanities stuff in (i) with the math stuff in (ii) in a coherent and plausible way. It's a lot to ask!

Since you have found Tolstoy's integration metaphor, I should probably mention that Stephen T. Ahearn wrote a 2005 article in the American Mathematical Monthly on this topic. (His article is freely available here.) Ahearn's article is quite thorough: I for instance would have a tough time trying to write a piece on this topic going beyond what he has already written. (And the fact that I've never read War and Piece is not exactly helping either...) If the assignment is "sufficiently serious", I would recommend that you pick some other integration metaphor to explain. (Exactly how one comes across "integration metaphors" is already not so clear to me, but the internet can do many magical things, probably including this...)

I should say though that in the United States at least it would be a very unusual calculus class that would require a student to complete such an assignment and be really serious about it, as above. (A part of me would really like to assign such an essay in my calculus class, but I think the results would be...disappointing.) If as you say the goal is to demonstrate knowledge of integration, then you should indeed concentrate on that. As ever, it couldn't hurt to talk to your instructor and get more specific information about this assignment: e.g. what is the suggested length of the essay? What sort of places does s/he have in mind for finding such a metaphor? Could you create your own metaphor? And so on.

In summary, if you put this question to us (at present the majority of the "answerers" are advanced mathematics students or math researchers) I fear you're setting yourself up to get picked on. It's probably best to clarify exactly what you need to do: it may not be so much, and it might just be worth taking a crack at it (as you've done) and seeing if that will be sufficient for the instructor.

P.S.: I have read some of Tolstoy's other works (especially Anna Karenina) and nothing math-related springs to mind. However, Dostoyevsky's Notes from Underground has some fun mathy material, although maybe not integration per se. I could imagine writing an ironic piece on whether integration (specifically, explicitly finding anti-derivatives) is as hard-scientific and deterministic as Dostoyevsky's view of mathematics is in this book, or whether the "art of finding antiderivatives" is messy and uncertain like the human condition. But, you know, this could be a failing essay!