Calculus: Analytic Geometry and Calculus, with Vectors

4.2 Riemann sums and integrals 213

subinterval so that to < tl < ti, let t2 be in the second subinterval so
that ti <= t2 < t2, and so on so that

(4.214) 4-1 < tk _< tk (1 < k S n).

Our machinery, which is still very much simpler than that in an elec-
trically operated dishwasher, enables us to produce numbers that are
called Riemann sums. We multiply f (tk ), the value of f at tk, by Ate,,
the length of the interval containing tk, and add the results. Thus, denot-
ing the Riemann sum by the symbol RS, we have

(4.22) RS = f (tl) At, + f (t2) f (t*) Ot3 +... + f (t*) Otn.

Because it takes too long to write this, we abbreviate it to the form

(4.221) RS = f(tk) Atk.

k-1

The right side is read "sigma k running from 1 to n eff of tee kay star

delta tee kay" and it denotes the sum of the terms obtained by giving k

the values 1, 2, 3, , n. The E (sigma) is called the summation

symbol, and it is very convenient.

Everybody should see that, when the function f and the numbers a

and x are given, it is easy to select the partition P in very many different

ways and to select the points tk in very many different ways. When an

electronic computer is kind enough to do the arithmetical chores, it is

even easy to produce very many Riemann sums.

Experience shows that we should avoid future difficulties by allowing

the partitions and Riemann sums to slumber peacefully while we invest

a moment to think about the names which we have attached to the parti-

tion points and the intermediate star points that determine them. The

points in Figure 4.212 were called to, tl, ,t.n and t', t2, ,t*.

We could, without changing the value of the Riemann sum, have called

these same points Xo, X1, ,X, and A*, XZ, , X. Thus there

is a sense in which the names of these points are "dummy names"; we

could have called the points is or u's or v's or X's or μ's or pi's. When

this matter is understood, we must ask and answer two questions. First,

why did we avoid the "natural" names xo, xi, , xand xi, x2*, ,

X,*? The answer is that we already have the interval from a to x on an

x axis appearing in our work, and we will have too many x's around the

house if we allow any more to enter. Secondly, why did we use the names

to, tl, ,t,a and tl, t*,. ,t*? The only answer we can give is

that they are as good as any and better than most alternatives. In

situations where we can conveniently use the "natural" names xo, xl,

, x,, and x*, x2, ,x*, we usually do so. Finally, we do not use

the letter i to denote "dummy integers" in (4.214) because the habit