[BUG] multiple integration

[Martin Jackson]

Part of the issue here is that we typically define single integrals 
in terms of coordinates using dx and define double integrals with the 
coordinate-free dA.  We could avoid or address some of the issues if 
we define single integrals essentially as line integrals using dl (or 
ds or whatever your favorite notation).  This has the advantage of 
producing more unified definitions:  For each, take the region, break 
it into pieces of length dl (or area dA), pick a point P in each 
piece, and sum the products f(P)dl (or f(P)dA).   Use \Delta l and 
index \Delta l and the points P if you want to display a sum and 
limit explicitly.   For straightforward interpretations of the sign 
of integrals, declare dl (or dA) to be positive as part of the 
definition (as Tevian proposes in his comments).  If f gives a 
density (charge or mass), then the integral of f gives the total 
(charge or mass).  When we introduce coordinates to evaluate an 
integral, we have to be careful with signs.  For f=1, the coordinate 
integral of f dx from x=2 to x=0 gives a negative value.  The 
physically meaningful quantity is the opposite of this negative value 
because dl=-dx here.



>I'm teaching multiple integration for the first time in several years.
>I'm emphasizing the idea that integrals are sums, among other things by the
>use of questions like:
>	Is the integral of y^3 dA positive, negative, or zero
>	over a region in the plane where y is negative?
>Two unexpected (to me) issues have arisen, one of which I feel is simply an
>error by the students, but the other is more subtle.
>
>The first issue is the desire of many students to integrate before answering
>the question, obtaining y^4/4, and arguing that this is positive, so the
>integral must be positive.  This is simply a mistake, but it does indicate how
>hard it is to get students to think about what integrals mean without
>calculating anything.
>
>The second issue is the subtle assumption that dA itself is positive.  This
>seems eminently reasonable when it really is an area, but is less obvious in
>contexts with different units.  More importantly, issues of orientation are
>being swept under the rug here.  One student explicitly integrated over all
>negative values of y, but integrated from 0 to minus infinity rather than the
>other way around.  Again, this is a mistake, but where is this taught?  The
>only good explanation I know involves surface integrals, where dA is the
>magnitude of the vector surface element, and hence contains an absolute value.
>
>Most texts hide this in the "definition" of multiple integrals as iterated
>integrals, which has an orientation built in.  The instructor can emphasize
>that the limits of such integrals always go from small values to large ones,
>but that can cause problems later when dealing with surface integrals, where
>there is no reason to make this assumption.  In fact, I think it's a bad idea
>to argue, as some mathematicians do, that one should always choose
>parameterizations of curves and surfaces so that the parameters increase;
>better to teach students to integrate from "start" to "finish" regardless of
>what that implies for the parameters.  But how then should one introduce this
>concept in multiple integration?
>
>I think the answer must be to emphasize that the area element dA really has an
>absolute value in it.  Whether this is done at the level of Riemann sums, as
>	delta A = |delta x| |delta y|
>or in differential notation simply as
>	dA = |dx| |dy|
>or simply in words doesn't really matter, so long as dA is explicitly assumed
>to be positive.
>
>Comments welcome.
>
>Tevian
>
>
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