1549901369-Elements_of_Real_Analysis__Denlinger_

7.2 The Riemann Integral Defined 367

Proof. (a) Suppose there exists a sequence {Pn} of partitions of [a, b] such

that S..(f, Pn)--+ L. By definition of J:f , Vn EN, S..(f, Pn) :==;J:f. Since limits

preserve inequalities (Theorem 2.3.lD lim S..(f, Pn) :==;J: f. That is, L :==;J: f.

n---too - -

(b) Exercise 8.

( c) Exercise 9. •

The next example shows how to use this sequential criterion in practice.

Example 7.2.13 Show that the function f(x) = x^2 is integrable on [O, l] and

find fol f.

Solution. Consider the sequence {Pn} of partitions of [O, l] given by
P n = { 0, ~, ~, · · · , ~}. Then Vi = 1, 2, · · · , n , 6.i = ~ and since f is increasing
on [O, l], mi = f (xi-1) and Mi= f(xi)· Then,

n n

(a) S(f, Pn) = L M i6.i = L f (x i )6.i

i=l i=l

= ~ ~ i^2 = ~ n(n + l)(^2 n + l) (See Exercise 1.3.4.)

n3 L n3 6

i=l

= ~ ( n : 1 ) ( 2n: 1 ) = ~ (^1 + ~) (^2 + ~) --+ ~.

• 1
  Thus, S(f, Pn) --+ 3--

(b) On the other hand,

n n

S..(f, Pn) = L mi6.i = L f (Xi-1)6.i

i=l i=l

n n-1

= ~ "°'(i - 1)2 = ~ """'j2

n3 L n3 L

i=l j=l

1 (n - l)n[2(n - 1) + l]

n^3 6

(See Exercise 1.3.4.)