1549901369-Elements_of_Real_Analysis__Denlinger_

366 Chapter 7 • T~e Riemann Integral

(b) Let 0 < c; < 1, and let Q = {O, 1-~, 3 + ~, 5}. Then Q is a partition
of [O, 5], and

S(f, Q) = M161 + M262 + M363

= 0 (1 - ~) + 1(2 + c:) + 0 (2 - ~)

= 2+c:.

0 1 3 5

t E \

1- 2 3 + f

Figure 7.4

Thus, since Ji f is the infimum of all the upper sums, f 05 f :::; S(f, Q) = 2 + c:.

Hence, Ve:> 0, Ji f:::; 2 + c:. Therefore, by the forcing principle, Ji f:::; 2.

(c) Putting (a) and (b) together with Theorem 7.2.7,

2 :::; Ji f :::; fo

5

f :::; 2.

That is, Ji f = f 05 f = 2. Therefore, f is integrable on [O, 5], and Ji f = 2. D

We now turn our attention to the problem of determining whether a given
function is integrable, and calculating the value of the integral. Sequences turn
out to be quite useful in this effort. The following theorem justifies a technique
often used in elementary calculus courses.

Theorem 7.2.12 (A Sequential Criterion for Integrability and Calcu-

lating J: f) Suppose f is defined and bounded on [a, b], and L E R

(a) If there exists a sequence {Pn} of partitions of[a, b] such that

$_(!, Pn) -+ L , then J: f ~ L.

(b) If there exists a sequence { Qn} of partitions of[a, b] such that

• b
  S(f, Qn)-+ M, then f a f:::; M.

( c) If there exist sequences {Pn} and { Qn} of partitions of [a, b] such that

$_(!, P n ) -+ L and S(f, Qn) -+ L , then f is integrable on [a, b] and J: f =

L.