write it down mathematically:
at the left end and call the (averaged) value of D for the first segment
D1. Call the first value of S, S1. The
product for the first segment is S1D1, and the area
under the product-curve is S1D1Dl.
The total area under that product-curve is the result of adding all N products:
+ . . . + SNDNDl
(= S SNDNDl)
The Greek sigma (S)
is used to indicate the sum without writing down all the terms.
Make the width of the red
segments smaller and smaller . . . and smaller yet...
and the sum become ever
smoother, the approximation ever closer to the exact value.
Sigma is a sharp-cornered
version of an "S" (for "sum").
When we switch to the smooth
summation, we change the sharp and irregular symbol to the elongated, smooth
version of "S," and we change the Greek D
(delta, for "difference") to a good old English "d." It looks like
This is, of course what Isaac
Newton invented for doing mathematics of this sort. This is part
of calculus, and the elongated "S" is an integral sign. The concept
of the smooth sum, the limit of extrapolating to infinitesimally small
widths of the red segments, is a simple concept. It's a simple concept
that's worth trying to get a good grasp of, because it's a little thing
that goes a long way to prepare a person for life in an age of modern science.
The complex part of a calculus
course is learning how to do the calculations. Today, we use the
computer to do most of the hard and tedious work.
Something else from mathematics
slipped in here. And it is considerably more sophisticated than calculus.