Math notation for integrals

The following is a revised version of a message I sent to Ron Whitney upon
receiving his about integrals.  I now see I should have posted to the list
at once.  I use a sort of TeX-influenced pseudo-notation below (if a
pseudo-notation is possible) to state examples.  If that is not good enough
please say so.

I guess that the impulse to use
  integral from ... to ... of ... \d x
comes from a feeling for templates with slots and from the speech
orientation.  It certainly is a basic (common) construction.  A friend and
I, exchanging formal math notation 30 years ago, used
Int[lower_limit, upper_limit, integrand, integrator]
notation, which it turned out was not unlike Mathematica's.  Now one can
wonder whether the integrator is to be a measure, say  $\d x$ (where I used
a \d for some letter d in suitable form) or $d\mu$, or a density (\`a la
Stieltjes), say $x$ or $g(x)$; or should one just allow anything, to take
care of $e^{-x^2/2} d\,x/s\pi$ and much more complicated things?

In any case, iterated integrals, if well nested, will work in this way as, say,

Int[l,u,Int[l1,u1,f(x,y), dx],dy]

and a special contour integral can be
CInt[\Gamma, f(x), dz/2\pi]

and a multiple one maybe

3Int[D\subset\reals^3,Q(x,y,z), d\, \vol]

or a path integral

Int[\scrS,,F(\omega) e^{iS(\omega), D(\omega)]

and so on ad nauseam.

There is the matter of the Jackson integral, which is a type of sum now
enjoying prominence, sometimes denoted with a heavy large sans-serif S, but
often with a conventional integral sign, perhaps with a subscript q to
denote the base of the numbers involved; and also Donsker's flat integral,
which uses a sort of squared-off integral sign; or then the Cauchy
principal value integrals which often have "P.V." before the integral sign
(or "P.P." if they are in French); etc. etc.

All these remarks seem to be in harmony with what Dave said.   His
statement about extendibility seems to be the most important new thing, and
we must all hope it can be implemented.

That there are integrals without limits $\int f$ or just written with the
measure as $\int f d\,\mu$ or $\int f \mu$ can surely be taken care of by
leaving out limits or an integrator.  Actually this notation moves toward
the understanding of integrals as linear functionals, then written $\mu(f)$
say, or the notion of integrals as setting up dualities between spaces.
But that passes on to more theory.  Standard notations for stochastic
integrals would be nice to support because of their use in business and
physics.

However, what we presumably are looking for is a cluster of constructions
that will go with the use of the Riemann integral in High School and
College math and that will allow later parsing to
1. present on a screen conventional notation
2. present in audio in a refined form of the common manner
3. allow, with optional hints added for the extra parsing, conversation with
symbolic processors for which the notation is
  Mathematica (p.812 of 2nd edition of Mathematica book):
  Integrate[f,x]
  Integrate[f,{xmin, xmax}]
  Integrate[f,{xmin, xmax},{ymin, ymax}]

OR
  Maple:  ...

OR
  Axiom (p.653 of Axionm book):
  integral(expression, symbol)
  integral(expression, segmentBinding)
  integral(f,x)
  integral(f,x=a..b)
or
  integrate(expression)
  integrate(expression,variable, [,options])
  integrate(f)
  integrate(f,x=a..b [,noPole])
OR
  Macsyma, Reduce, .....

I think it is the drive to semanticize (?) that is the reason for the
change from TeX style.  This is just a name for passing hints to other
models in the present context, as far as I can see.  Interestingly, the two
examples from CA systems written out above show there is a matter of
distinction between Integral as integration and integral as in integers (or
integral domains, integral lattices, ...) which one enforced by convention.


How do others view this relatively simple collection of examples?

                Patrick