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Re: Isn't it the limit?





On Sun, 30 Mar 1997, John Mallinckrodt wrote:

Don has asked us how we would explain to a student why we do not correctly
obtain the area of a sphere by adding the areas of what naively appear to
be infinitesimal triangles with vertices at a pole and infinitesimal bases
along the equator. This is a really good question as it concerns one of
the critical elements in setting up integrals which is, in turn, one of
the hallmarks abilities of a good physicist.

Clearly, as has already been discussed, the reason the "triangle method"
doesn't work is that the "triangles" aren't triangles. Nevertheless,
understanding this still leaves the original question unanswered.

....

At any rate, I still have no particularly good answer to the original
question, but I do hope to stimulate more consideration of this
interesting and, I think, important question.


Thank you, John, for correctly interpreting my concern, and stating it
better than I did.

I've seen math and physics teachers and even some textbooks use the slab
and ribbon methods for finding volumes, and also for calculating first and
second moments, and never mentioning the pitfalls, nor clearly stating the
conditions under which the method is valid. Also, they don't attempt the
application of it in situations like this one where it fails, or where is
is so easily misapplied. There are many other situations in physics
teaching where we casually use methods without full explanation, and never
happen to consider those situations where difficulties arise.

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Prof. of Physics Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745 CIS: 73147,2166
Home page: http://www.lhup.edu/~dsimanek FAX: 717-893-2047
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