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Topic: Derivation for surface area of revolution
Replies: 7   Last Post: May 25, 2012 8:41 PM

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Jim Rockford

Posts: 164
Registered: 6/30/06
Re: Derivation for surface area of revolution
Posted: May 25, 2012 11:41 AM
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Thank you all for your replies, but I guess we're having a

First, I do not find the presentation of the derivation of the surface-
area-of-revolution formula in calculus books to be unclear or
otherwise poorly presented. I just find them lacking in their common
avoidance of addressing a very natural question that arises in the

Let me again try to explain why I find the responses I've read on this
thread similarly lacking. Yes, to some it may seem "obvious" that you
can't add up the surface area of infinitesimally thin cylinders (i.e.
using dx instead of ds), and that curvature needs to be respected
more explicitly. However, you could equally well make that
"intuitive" argument in the derivation of the formula for volumes of
revolution as well. Why does "dx" suffice (thence at first glance
completely ignoring curvature of the curve revolved) in that case but
not in the surface area formula? Obviously, it has something to do
with the difference between calculating volumes and areas. In some
sneakier way the curvature is accounted for in the volume calculation
by the cross-sectional areas being added up. But this is never
clearly explicated in calculus books, nor have I seen here anything
except standard "intuitive" arguments (and even that you won't find in
today's name-brand calculus books). Perhaps you all don't find these
arguments lacking, but I do. They're certainly not suitable for
serious, inquisitive students. They're good having waving
explanations, but that's all, in my opinion (note: I don't mean to be
overly critical, as I don't have a better way to explain this either!)

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