22, 23] DEFINITE INTEGRALS. 39 



But in this sum P, for every s the greatest possible difference 

 fsf"n+k is in absolute value not greater than the oscillation D 8 . 

 Therefore 



In like manner 



where [jPJs /!>/. 



Then 2 S r /^S ,'/,' -PP, 



1 1 



n n 



and if lim 2 8 S D 8 = for all systems of division, the limit 2 S s / g 



tt=co 1 1 



is the same for all systems of division. It is easy to show that if 



n 



the condition lirn 2 B s D s = is satisfied for one mode of division, 



it is satisfied for all. This is then the necessary and sufficient 

 condition that the function f(x) shall be integrable in the interval 

 ab. 



23. Properties of Definite Integrals. It results imme- 

 diately from the definition 



b n 



f(x) dx lim 2 &sfs> 



i n=<x> 1 



that if we interchange the limits a, b, since every S s changes sign, 

 the sign of the integral is changed. More generally 



(I) 



J a J b 



The arithmetical mean of a number of quantities is defined as 

 their sum divided by their number. If f(x) is finite and in- 

 tegrable in an interval ab, and 8 3 8 n , 8/ n > are two 



divisions of the interval, from the last equation of 22, 



is v/;- 1; ,/.! s i s 8 D 8 +l s; A'. 



1111 



Consider n constant and let n' increase without limit. 

 Then 



u, 



f(x) dx-2 B s f s \%8 8 D 8 , so that I B s f s 



