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!
 A: 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!  
A: I think that this is a really great assignment, and I think
that you likely have an unusual and enlightened instructor,
from whom I would encourage you to try to learn as much as
you can.
If I were to undertake this assignment, however, I would
adopt a more critical tone about the strength of the
metaphor. Tolstoy is saying that the movement of humanity
arises as the continuous sum of each individual's
infinitesimal contribution to it, and is continuous as a
result, just as the area under a curve can be thought of as
the sum of the increasingly large numbers of increasingly
thin rectangles below it, each contributing infinitesimally
to the area. Perhaps such a perspective would lead one to a
morose attitude on the fate of humanity and the ability of
of an individual to affect it---after all, if your
contribution has only an infinitesimal affect, then you
will not significantly change the outcome.
But my personal outlook on life would compel me to resist
this perhaps-depressing conclusion. And so for such an assignment as you have described, I
would search for mathematical grounds on which to do so.
First of all, of course, it seems that the development of
humanity is not continuous; it is rather punctuated by
singular developments, such as scientific advances and
discovery or political events, such as revolution. In
addition, from our study of history it often seems that
certain individuals can have and often have had a
non-infinitesimal affect on human progress. Think of the
great inventors and innovators in history, who changed the
course of humanity and scientific development. Isaac
Newton was not a slender thin rectangle, contributing only
infinitesimally to humanity. And neither were the other
great thinkers in our history. (And perhaps consider also
those who have had a great negative affect.) Although
these individuals may not have acted alone, but made their
critical actions after the actions of many others that came
before, perhaps they stood on the shoulders of giants---and
this will be Tolstoy's reply---the counterargument is that
nevertheless it was often comparatively small groups that
led to outsize developments, and so their contributions
could not have been infinitesimal. After all, the sum of
finitely many infinitesimals is still infinitesimal, so if
a small group has had outsize total affect, the individual
contributions must have been non-trivial. Finally, since there are
indeed only finitely many people altogether, the effect of
each of them will not be an infinitesimal proportion of the
total effect.
I often make term-paper writing assignments for my more
advanced courses (and almost always in my graduate
courses), but I am now inspired by your question to look
for ways to have my calculus-level students undertake more
writing.
A: the drop and bucket problem and economics. each person contributes nothing indiidually to the ecnomy. but together as a whole they do.
