Calculus: Analytic Geometry and Calculus, with Vectors

(^212) Integrals
solve one part, pick an integral formula from a (preferably your) bool, of tables
that has the form
f f (x) dx = F(x) + c
(except that most tables omit the constants) and show that F'(x) = f(x). This
promotes an understanding of integral formulas and provides practice in differ-
entiation. It is never too early to start acquaintance with formulas involving
X where X is one or another of a + bx, a2 + x2, a2 - x2, ax2 + bx + c, etcetera.
In these situations, modesty and timidity are not virtues. We profit most when
we attack the problems that seem most impenetrable and discover that they
really are very simple.

4.2 Riemann sums and integrals

and integrals that are named after Riemann (1826-1866) in spite of the

fact that Archimedes (287-212 a.e.) knew how special ones could be

used in a few special cases. Let f be a function which is defined over an

interval a <- x < b and has values f(x) such that

(4.21) m<f(x) <M (a<x5b)

where m and M are constants. This amounts to saying that f is bounded

over the interval; M is an upper bound and m is a lower bound. Our next

few steps are so simple that it may be difficult to see why they are impor-

tant. As in Figure 4.212, let x be a fixed (or selected) number for which

a < x 5 b. Thus x can be b, but it is not necessarily so. Let n be a

positive integer. We make a partition P of the interval from a tox

into n subintervals by inserting points to, t1, t2j , t.-I, tn, where

(4.211) a=to<tj<t2< <tn_1<tn=x.

These points are the circled points of the figure and are the end points

of the subintervals.

O

tl

O

t t3a

to tl t2 t3 -0tA-11 ti Q

t,,_2 to-1 t

Figure 4.212

b

t

Let At, denote the length of the first subintervalso that At, = ti - to,

let Ot2 denote the length of the second subinterval so that At2= t2 - ill

and so on so that

(4.213) Atk = tk - tk-1 (1 5 k 5 n).

It is not required that the points to, t1, , in be equally spaced. The

greatest of the numbers Ail, Ot2j - - , Atn is called the norm of the parti-

tion P and is denoted by the symbol IPJ. Thus JPJ is the length of the

longest of the subintervals in P. Our next act introduces the star char-

acters. Let tl (read tee one star) be a number (or point) in the first