1549901369-Elements_of_Real_Analysis__Denlinger_

374 Chapter 7 • The Riemann Integral

Adding together inequalities (3) and (4), we have

• 
  
  
  
  • E
    S(f, Q) - S.(f, Q) < S(f, P) - S.(f, P) + 2
    E E
    <
    2
  
  
  
  

+

2

[from(l) above]

= E.

Therefore, 'llt: > 0, 3 5 > 0 3 'II partitions Q of [a, b], 11 Qll < 5 ::::} S(f, Q) -
8_(f, Q) < E. •

THE INTEGRAL VIA RIEMANN SUMS

While Darboux sums lead quite effectively to a natural definition of J: f
there are certain advant ages to be gained from a slightly different type of sum
called a "Riemann sum." This type of sum will allow us to characterize the
integral as a certain type of limit. It will also serve as a basis for developing
numerical integration techniques in other courses (not this one).

Definition 7.3.3 A tagged partition P of [a, b] is a partition
P = { xo, x 1 , x2, · · · , Xn} of [a, b] along with a set of "tags" x; E [xi-l, xi] for
n
i = 1, 2, · · · , n. Then the sum R(f, P) = L f(x7)6i is called the Riemann

sum of f over P*.

y

a=xo:

xj

i=l

y =f(x)

x

X1 X2 Xi-I i /i

X2 X;

The shaded area represents R(f, P*)

Figure 7.5