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**Unformatted text preview: **CHAPTER (a, 0) FIGURE 1 FIGURE 2 INTEGRALS The derivative does not display its full strength until allied with the “integral,”
the second main concept of Part III. At ﬁrst this topic may seem to be a com-
plete digression—in this chapter derivatives do not appear even once! The
study of integrals does require a long preparation, but once this preliminary
work has been completed, integrals will be an invaluable tool for creating new
functions, and the derivative will reappear in Chapter 14, more powerful than
ever. Although ultimately to be deﬁned in a quite complicated way, the integral
formalizes a simple, intuitive concept—that of area. By now it should come
as no surprise to learn that the deﬁnition of an intuitive concept can present
great difﬁculties—“area” is certainly no exception. In elementary geometry, formulas are derived for the areas of many plane
ﬁgures, but a little reﬂection shows that an acceptable deﬁnition of area is
seldom given. The area of a region is sometimes deﬁned as the number of
squares, with sides of length l, which ﬁt in the region. But this deﬁnition is
hopelessly inadequate for any but the simplest regions. For example, a circle
of radius 1 supposedly has as area the irrational number 71', but it is not at all
clear what “71' squares” means. Even if we consider a circle of radius 1/\/7r,
which supposedly has area 1, it is hard to say in what way a unit square ﬁts in
this circle, since it does not seem possible to divide the unit square into pieces
which can be arranged to form a circle. In this chapter we will only try to deﬁne the area of some very special
regions (Figure 1)~those which are bounded by the horizontal axis, the verti-
cal lines through (a, 0) and (b, 0), and the graph of a function f such that
f(x) 2 0 for all x in [0, b]. It is convenient to indicate this region by R(f, a, b).
Notice that these regions include rectangles and triangles, as well as many
other important geometric ﬁgures. The number which we will eventually assign as the area ofR(f, a, b) will be
called the integral of f on [a, 1)]. Actually, the integral will be deﬁned even for
functions f which do not satisfy the condition f(x) 2 0 for all x in [1.1, b]. Iff is
the function graphed in Figure 2, the integral will represent the difference of
the area of the lightly shaded region and the area of the heavily shaded
region (the “algebraic area” of R(f, a, 17)). The idea behind the prospective deﬁnition is indicated in Figure 3. The
interval [51, b] has been divided into four subintervals U0: til [tn t2] [[2, is] [13, t4]
by means of numbers to, t1, t2, 33, £4 with a=to<l1<t2<l3<t4=b
214 FIGURE 3 DEFINITION DEFINITION Integrals 215 (the numbering of the subscripts begins with 0 so that the largest subscript
will equal the number of subintervals). On the ﬁrst interval [150, t1] the function f has the minimum value m1 and the
maximum value M1; similarly, on the ith interval [t,-_1, ti] let the minimum
value off be m,» and let the maximum value be Mi. The sum 3 = m‘(t1 — to) + 7712(62 "‘ t1) + "1303 — 52) + ”1404 —‘ 53) represents the total area of rectangles lying inside the region R(f, a, 1)), while
the sum S = M1051 ‘ to) ‘l‘ M202 — £1) + M303 — t2) + M404 — l3) represents the total area of rectangles containing the region R(f, a, b). The
guiding principle of our attempt to deﬁne the area A of R( f, a, b) is the obser-
vation that A should satisfy 53A and ASS, and that this should be true, no matter how the interval [(1, b] is subdivided. It is to
be hoped that these requirements will determine A. The following deﬁnitions
begin to formalize, and eliminate some of the implicit assumptions in, this
discussion. Let a < b. A partition of the interval [0, b] is a ﬁnite collection of points in
[(2, b], one of which is a, and one of which is b. L. The points in a partition can be numbered to, . . . , 25,, so that
a=lo<t1< ‘ ' ' <tn_1<tn=b; we shall always assume that such a numbering has been assigned. Suppose f is bounded on [(1, b] and P = {to, . . . , tn} is a partition of
[51, 1)]. Let m, = inf {f(x): ti_1 S x 5 1,}, M,- = sup {f(x): ti—l S x S 5‘}- The lower sum of f for P, denoted by L( f, P), is deﬁned as LMH=2mm—my The upper sum off for P, denoted by U (f, P), is deﬁned as UMH=2MM—md The lower and upper sums correspond to the sums s and S in the previous 216 Derivatives and Integrals example; they are supposed to represent the total areas of rectangles lying
below and above the graph off. Notice, however, that despite the geometric
motivation, these sums have been deﬁned precisely Without any appeal to a
concept of “area.” Two details of the deﬁnition deserve comment. The requirement that f be
bounded on [(2, b] is essential in order that all the m and M.- be deﬁned. Note,
also, that it was necessary to deﬁne the numbers m.- and M.- as inf’s and sup’s,
rather than as minima and maxima, since f was not assumed continuous. One thing is clear about lower and upper sums: If P is any partition, then L(f, P) S U(f, P), because 71. LMH=2mm—m% i=1 Umm=jmeﬁm, and for each i we have
mi<£i " kg) S MM; — l‘i—1)- FIGURE 6 Supfiose ﬁrst that f(x) = c foi' all x in [a, b] (Figure 6). If P = {A}, . . is any partition of [0, b], then
m,- = M; = C,
so 11 ML P) = Z w.- — m) = c<b — a), i=1 U0: P) = Z c<z.- — :H) = ca: — a). '=1
In this case, all lower sums and upper sums are equal, and
sup {L(f: P)} = inf {U(f, Pl} = 6(1) - (I)-
Now consider (Figure 7) the function f deﬁned by O x irrational
f(x) = { ’ .
1, x ratlonal. .,¢,.} Integrals 219 If P = {10, . . . , in} is any partition, then m,- = 0, since there is an irrational number in [n-1, ti],
and M, = 1, since there is a rational number in [t,~_1, ti].
Therefore, 71. uﬁn=20wwwm=m UMB=Elww+J=b—m =1 FIGURE 7 Thus, in this case it is certainly not true that sup {L(f, P)} = inf {U(f, P) }.
The principle upon which the deﬁnition of area was to be based provides
insufﬁcient information to determine a speciﬁc area for R( f, a, b)—any num-
ber between 0 and b —- a seems equally good. On the other hand, the region
R(f, a, b) is so weird that we might with justice refuse to assign it any area at
all. In fact, we can maintain, more generally, that whenever SUP lL(f, P)} ¢ inf {U(f, 13)}, the region R( f, a, b) is too unreasonable to deserve having an area. As our
appeal to the word “unreasonable” suggests, we are about to cloak our
ignorance in terminology. DEFINITION A function f which is bounded on [[1, b] is integrable on [(1, b] if
sup {L(f, P): P a partition of [0, b]} = inf {U(f, P): P a partition of [(1, b]}. In this case, this common number is called the integral of f on [(1, b] and
is denoted by
b
L f- (The symbol f is called an integral sign and was originally an elongated s, for
“sum;” the numbers a and b are called the lower and upper limits of integration.) The integral [abf is also called the area of R(f, a, b) when f(x) 2 0 for all x
in [52, b]. If f is integrable, then according to this deﬁnition,
L( f, P) g [0” f g U(f, P) for all partitions P of [(1, b]. b . . . .
Moreover, f f 15 the unique number With this property. This deﬁnition merely pinpoints, and does not solve, the problem discussed
before: we do not know which functions are integrable (nor do we know how
to ﬁnd the integral off on [a, b] when f is integrable). At present we know only 220 Derivatives and Integrals THEOREM 2 PROOF two examples:
(1) iff(x) = c, then fis integrable on [(1, b] and [ff = c - (b -- a).
(Notice that this integral assigns the expected area to a rectangle.) (2) iff(x) = 0, xirrational 1, x rational, then f is not integrable on [a, b]. Several more examples will be given before discussing these problems fur-
ther. Even for these examples, however, it helps to have the following simple
criterion for integrability stated explicitly. Iff is bounded on [a, b], then f is integrable on [(1, 1)] if and only if for every
5 > 0 there is a partition P of [(2, (2] such that U<faP)_L(f3P)<E Suppose ﬁrst that for every 5 > 0 there is a partition P with Ulf, P) — L(f, P) < 6- Since
inf {U(f, P’)} 3 ML P), SUP {1405199} 2 Llf, P),
it follows that inf {U02 P’)} - SUP {LUZ P’)} < 8- Since this is true for all e > 0, it follows that SUP {L(f, P’)} = inf {U(f, P')}; by deﬁnition, then, f is integrable. The proof of the converse assertion is simi-
lar. If f is integrable, then SUP {Llﬂ PM = inf {Ulﬁ P)}- This means that for each 8 > 0 there are partitions P’, P" with
(17(f3 PH) _ L(fs PI) < 5. Let P be a partition which contains both P' and P”. Then, according to the
lemma, Ulf, P) S U(f, P”),
LU, P) 2 UL P');
consequently, U(f9-P) _L(f,P)S U(f:P,l)—L(f3P,) <E'I Although the mechanics of the proof take up a little space, it should be clear
that Theorem 2 amounts to nothing more than a restatement of the deﬁnition Integrals 221 of integrability. Nevertheless, it is a very convenient restatement because there
is no mention of sup’s and inf’s, which are often difficult to work with. The ...

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- Fall '14
- PARKINSON,J
- Microeconomics