Calculus: Analytic Geometry and Calculus, with Vectors

214 Integrals

of using i leads to awkwardness when we finish study of calculus and enter

realms where i is always the imaginary unit whose square is -1. We

use k because it is as good as any and better than most.

We now come to the most fundamental remark that appears in the

theory of Riemann integration; analogous remarks appear in theories

of other integrals. Depending upon the function f and the numbers

a and x that have been selected, it may be true(or it may be false) that

there is a number I such that to each positive number e there corresponds

a positive number S such that

n

(4.23) 1 f(tk) Litk - II < e

k=1

whenever JPJ < S. This is, of course, just a precise way of saying that

there may be a number I such that each Riemann sum with a small norm

is near I. If this I exists, then f is said to be Riemann integrable over the

interval from a to x and I is said to be the Riemann integral off over the

interval. This integral is denoted by the symbol in the formula

(4.24) I= fXf(t)dt

and the symbol is read "the integral from a to x of eff of tee dee tee."

The numbers a and x are called the lower limit and the upper limit of

integration, and we always read the lower one first. The symbol t is

called a dummy variable of integration, the derogatory terminology being

applied because the value of the integral would be the same if t were

replaced by s or u or a or 0 or any other symbol that cannot be confused

with a, x, f, and d. It is a convenience (and sometimes also a source of

misunderstanding, confusion, and controversy) to drag in the notation

of limits and write '

(4.25) lim f (tk) Litk = f a5 f (t) dt.

jPj-'O k=1

A much more substantial convenience results from boiling this down to

(4.251) lim I f (t) At = f ''f(t) dt,

the idea being that we can restore the omitted embellishments whenever

there is a reason for doing so.

In case no such number I exists, we say that f is not Riemann integrable

over the interval from a to x and that faZ f (t) dt does not exist (that is,

does not exist as a Riemann integral). To emphasize the fact that a

bounded function f and an interval a < x S b can be such that af b f (t) dt