CHAPTER III. 



DEFINITE INTEGRALS. 



21. Definite Integral of a Function of one Variable. 



If we consider a continuous function of one real variable, the 

 notion of its definite integral may be illustrated by means of a 

 geometrical representation. If the function y =/(#) be represented 

 as the ordinate of a curve of which x is the abscissa, and if between 

 two points x = a, x b, we place any number n 1 of points 

 #1, # 2 , ...... ofc, ...... #n-i> and in the intervals between them erect 



ordinates to the curve at points % lt | 2 ...... so that 



the sum 



S = (a, - a)/() + (x, - 

 represents the area of the rectangles constructed on the bases 



with the altitudes /(&) The value of this sum depends on the 

 form of the curve or of the function f(x\ on the choice of the 

 points of division, x ...... a? n , and of the points f& within the 



intervals. It can be shown, however, that if all the differences B k 

 are less than a certain value S, all the values that S can take are 

 confined between certain limits, and if the number of intervals 

 increases so that 8 decreases without limit while S x + & 2 ...... + & n 



remains always equal to b a, that these extreme values of S 

 approach a common limit. This limit will represent the area of 

 the space bounded by the axis of X, the ordinates erected at the 

 points x = a and x = 6, and the curve representing the function 



y =/() 



