1549901369-Elements_of_Real_Analysis__Denlinger_

7.3 The Integral as a Limit of Riemann Sums 377

THE INTEGRAL AS A LIMIT

Note: Because of Theorem 7.3.5 it makes sense to write

n

t f = lim R(f, P*) = lim "°""' f (x:)6.i.

a ll'P* 11-->0 llP* 11-->0 L.., i=l

(if this limit exists and is independent of the tags xi)

It is important to note that this is a new kind of limit, not covered by any of
our previous definitions of limits. Its definition is given in the boxed statement
in Theorem 7.3.5.
Using the limit criterion for integrability, we can develop a very practical

method of using sequences to calculate l: f if we know that f is integrable on
[a,b].

Theorem 7.3.6 (Sequential Limits for Calculating l: f)
Suppose f is integrable on [a, b], and {Pn} is a sequence of partitions of [a, b]
such that llPnll __, 0. Then

(a) S..(f, Pn) , l: f and S(f, Pn) , l: f ·

(b) If each P; is tagged, then R(f, P;) --. l: f , regardless of the choice of

the xis.

Proof. (a) Suppose f is integrable over [a, b]. Let E > 0. By the Rie-
mann/Darboux criterion for integrability, j 8 > 0 3

llPll < 8 =* S(f, P) - S..(f, P) < c.

Since llPnll __, 0, ::lno EN 3 n :2: no=* llPnll < 8. Thus,

n :2: no :::? S(f, Pn) -5_(!, Pn) < E. (6)

Recall from the "equivalent form" of Riemann's criterion for integrability

(7.2.15) that there is a unique number I ( = l: f) between all upper and lower

sums. Thus, we can write (6) as

n :2: no :::? [S(f, Pn) - I] + [I -S..(f, Pn)] < E

:::? 0 :::; S(f, Pn) - I < E and 0 :::; I - S..(f, Pn) < E.

Therefore, S(f, Pn) --. I and S..(f, Pn) __, I. But I= l: f.

(b) Exercise 6. •