Calculus: Analytic Geometry and Calculus, with Vectors

250 Integrals

and notations have their historical origins in primitive ideas that are fuzzy or

incorrect. The number in the right member of the formula

(1) lim I f(x) Ax = fa f(x) dx

is, when it exists, defined in terms of Riemann sums in a way which we must

now understand. If (1) holds, then to each positive number e there corresponds

a positive number S such that

(2) I f(xk) Axk -fa b f (x) dx I< E

km1

whenever P is a partition of the interval a 5 x < b for which JPJ < S. For a

long time before this precise idea of Riemann revolutionized (or counter-revolu-

tionized) mathematics, it was generally considered to be meaningful to regard

the limit of sums as "the sum of infinitely many infinitesimals." Thus faaf(x) dx

was considered to be an "infinite sum" of products of "finite" numbers f(x) and

"infinitesimal" numbers dx. The "reasoning" involved is quite as flimsy and

unrewarding as the "reasoning" which reaches the "conclusion" that "a circle

is a polygon having infinitely many infinitesimal sides because it is a limit of

polygons." In mathematics, as in other sciences, many of our ancestors were

intrigued by ideas which are now considered to be obsolete. Nowadays we accept

the idea that the sum of the volumes of many thin slabs can be a good approxima-

tion to the volume of a spherical ball, but we reject the fuzzy idea that the

volume of the ball is the sum of the volumes of infinitely many infinitely thin

slabs. It is not easy for historians to decide which of our great ancestors really

had quite correct ideas about approximations and limits and, without swallowing

ideas about sums of infinitesimals, merely used the fuzzy terminology because it

was the fashion to do so. There can be tenuous connections between ideas and

words. If Leonhard Euler wrote in a language in which apples were called

"potatoes that grow in the air," historians unaware of the fact have an oppor-

tunity to conclude that this intellectual giant did not know the difference between

potatoes and apples. Some people believe that the notation for integrals is bad

because it makes too many people think that the dx is a number. The author

believes that terminologies and notations involving limits are the real sinners

because they make too many people think that numbers and partitions and other

things are mobile. Perhaps replacing "lim" by "approx" in (1) would cure many

of our ills.

4.6 Riemann-Cauchy integrals and work This section introduces

integrals that are, in some cases, not Riemann integrals but are con-

structed from Riemann integrals by use of ideas that were made precise

by the French mathematician Cauchy (1789-1857). It may happen that

the integral in the right member of the formula

(4.61) f 'f(x) dx= limf 'f(x) dx

h-> m a