1549901369-Elements_of_Real_Analysis__Denlinger_

7.3 The Integral as a Limit of Riemann Sums 379

Definition 7.3.8 Suppose f is defined and bounded on [a, b], where a < b.

For each n E N, the regular n-partition of [a, b] is the partition Qn

{xo,x1,x2,·· · ,xn} of [a,b] into n subintervals of equal length 6 = llQnll =

b-a

--. That is , Xi = a+ i6 for i = 0, 1, 2, · · · , n.

n

In this case, the upper and lower Darboux sums simplify somewhat:

n n

$._(!, Qn) = 6 L mi and S(f, Qn) = 6 L Mi

i=l i=l

n

and thus, S(f, Qn) - $._(!, Qn) = 6 I: (Mi - mi)· •

i=l

In elementary calculus courses, regular partitions are usually preferred over
the more general partitions we have been using here, because in a first encounter
they seem easier to understand and calculate. In fact, many elementary calculus
courses use regular partitions exclusively. You probably noticed that we used
them in finding J 0
1
x^2 dx in Example 7.2. 13 , in finding J 14 (x^2 - 4x + 5)dx in Ex-
ample 7.3.7, and in proving that monotone functions and continuous functions
are integrable (Theorems 7.2.16 and 7.2.17).
The ability to use general partitions is an important tool in the analyst's
tool kit, both for specific calculations, as in Example 7.2. 11 , and for proving
general theorems. Nevertheless, many students and their instructors often ask
whether the exclusive use of regular partitions in the definition of Riemann inte-
grability can be rigorously justified. Since this issue is virtually never discussed
in calculus textbooks, the answer is not well known. The following theorem
suggests that the answer might be "yes."

Theorem 7.3.9 Suppose f is defined and bounded on [a, b], where a < b. Let
{ Qn} denote the sequence of regular n-partitions of [a, b]. Then f is integrable
over [a, b] {:} both sequences {$._(!, Qn)} and {S(f, Qn)} converge, and have the

same limit. In this case, t a f = n~oo lim $._(!, Qn) = n~oo lim S(f, Qn).

Proof. To prove the '* direction, apply Theorem 7.3.6. To prove the ~
direction, apply Theorem 7.2. 12 (c). •

Remarks 7.3.10 It is tempting to believe that when f is integrable over [a, b],
the sequence {$._(!, Qn)} of lower sums for the regular n-partitions of [a, b] is
monotone increasing, and the sequence {S(f, Qn)} of upper sums is monotone
decreasing. This is not necessarily true, even though both sequences converge
to J: f. The reason why these sequences are not necessarily monotone is that
the partition Qn+i is not a refinement of Qn when n > l. A counterexample
is given in Exercise 12.