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Re: integral nomenclature

While I agree with your aargh!!!, and appreciate John Denker's
comment, I can see where this casual view of the measure arises. Most
calculus texts (and the one I am writing is no exception) introduce the
integral as the antiderivative. And it is introduced at the stage where
the student is dealing only with 1-dimensional problems. We are
concerned, then, with a pair of functions of a single variable: F(x) and
its derivative f(x). At this level there is no ambiguity about writing
the pair as F(x) = \int f(x), especially when the indefinite integral is
intended. But such practice is short-sighted and conducive to the
development of bad habits and generally immoral behavior, IMO.
Part of the cure is to start early with the derivative as a ratio:
f(x)=dF/dx. The inverse operation then consists of two steps. First is
dF = f(x)dx, simple algebra, and each side is an infinitesimal. The
second step is "grow" the infinitesimals with \int dF = F =\int f(x)dx.
This approach is not favored by math educators who don't like to
view the derivative as a ratio (I think because of the limit process
involved). That is one of the ways that my textbook differs from most
calculus texts, since I don't rely on the notion of limit for the
derivative (see the introduction to my text at
www.hep.anl.gov/jlu/index.html).
Regards,
Jack


On Fri, 21 Feb 2003, Carl E. Mungan wrote:

Consider the integral Int{f(x) dx}. As physicists, probably many of
us read and think of this as a summation of quantity f multiplied by
small intervals dx (where f is evaluated say in the middle of each
interval for specificity). We note that dx is dimensional and carries
units (typically). But I'm sure many of us have experience with
students leaving off the dx, which doesn't seem all that important to
them, until suddenly f(x) is not just some mathematical function like
2x+5 but a real physical quantity with say charges, angles,
positions, and other things, more than one of which may be varying
(and thus first need to be related to each other before the integral
can be evaluated).

Anyhow, this all seems rather basic. However... in frustration I
looked in a typical intro calculus text (the one I used as a
freshman) to see how this is first presented and the section which
introduces the integral has a subsection titled "Terminology and
Notation" and are you ready for this? I'll loosely quote but without
changing what the text is clearly saying:

"The integral is denoted Int{f(x) dx} or Int{f}. The symbol Int is an
integral sign. It is a stylized S to suggest the connection with
sums. f is called the integrand. The symbol dx is a device for
keeping track of the variable."

AARGH #1: The text appears to condone the dropping of dx.
AARGH #2: dx is not even given a name. It's just a "device."

Okay I've calmed down. But now seriously:

1. What would you name "dx" in an integral specifically? I sometimes
call it the "differential" but this makes more sense in connection
with derivative, DE's, and such.

2. What advice have you gained in your years of teaching physics that
help students understand that "dx" is not something you can insert or
remove from an integral on an as-needed basis?

Helpful comments about using integrals with introductory college
students are welcome, Carl
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu http://physics.usna.edu/physics/faculty/mungan/


--
"What did Barrow's lectures contain? Bourbaki writes with some
scorn that in his book in a hundred pages of the text there are about 180
drawings. (Concerning Bourbaki's books it can be said that in a thousand
pages there is not one drawing, and it is not at all clear which is
worse.)"
V. I. Arnol'd in
Huygens & Barrow, Newton & Hooke