21, 22] DEFINITE INTEGRALS, 37 



This conception may be extended to any function whether 

 continuous or not, and the limit, if there be any, approached by the 

 sum 



Urn S= Km *lf $*/(&), 



n = <x> n = <x>k=I 



as the number of intervals is made to increase without limit, is 

 the definition of the definite integral of the function f(x) from a 

 to b. It is denoted by 



"b 



f(x) dx, 



L 



f(x) is called the integrand, a and b the limits, and ab the field of 

 integration. Evidently the letter x in the symbol may be replaced 

 by any other without affecting the integral. If the sum has a 

 limit the function / (x) is said to be integrable in the region from 

 a to b. 



22. Condition of Integrability. The oscillation of a func- 

 tion in a given interval is the difference between the greatest and 

 the least value that it assumes in that interval. It is evident 

 from the definition of continuity that if e is a positive number as 

 small as we please we may always find a number 8 such that in 

 any interval less than 8 and lying in the region ab in which the 

 function is continuous, the oscillation is less than e. 



Let fj ...... f n be a system of ordinates for a system of sub- 



division #! ...... x ny and let f / ...... f w ' be a different set of ordinates 



contained in the same intervals 8 1} 8 2 ...... 8 n . 



Then 



I ./(.) - 1 8./(.') = 28. </(.) -/(&')). 



Then we may find 8 so that when all 8 s 's become less than 8, 

 every 



I /<&) - 

 and consequently 



) < 2 S s e = (6 - a) e. 



1 1 



As n increases indefinitely, e decreases indefinitely, and 



so that the selection of the ordinates in the intervals does not 

 affect the limit, if one exists. 



