SECT. IV.] AREAS REPRESENTING INTEGRALS. 427 



whatever positive number p may be. If we suppose, for example, 



1A ,, T ,. .sn , 



p = 10, the curve whose ordmate is - - has sinuosities very 



x J 



much closer and shorter than the sinuosities whose ordinate is 

 ; but the whole area from x = up to x = x is the same. 



x 



Suppose now that the number p becomes greater and greater, 

 and that it increases without limit, that is to say, becomes infinite. 



The sinuosities of the curve whose ordinate is - are infinitely 



ss 



near. Their base is an infinitely small length equal to - . That 

 being so, if we compare the positive area which rests on one 

 of these intervals -- with the negative area which rests on the 



following interval, and if we denote by JTthe finite and sufficiently 

 large abscissa which answers to the beginning of the first arc, 

 we see that the abscissa a?, which enters as a denominator into 



the expression of the ordinate, has no sensible variation in 



the double interval , which serves as the base of the two areas. 



Consequently the integral is the same as if x were a constant 

 quantity. It follows that the sum of the two areas which succeed 

 each other is nothing. 



The same is not the case w r hen the value of x is infinitely 



small, since the interval has in this case a finite ratio to the 



P 



r 01 p T?*/ 1 



value of x. We know from this that the integral / dx , in 



Jo * 



which we suppose^? to be an infinite number, is wholly formed out 

 of the sum of its first terms which correspond to extremely small 

 values of x. When the abscissa has a finite value X, the area 

 does not vary, since the parts which compose it destroy each other 

 two by two alternately. We express this result by writing 



x 



