40 DEFINITE INTEGRALS. [INT. III. 



is an approximate value of the integral whose error is less than 



We may put the B s 's all equal, so that S s = - . Then 



(2) - f(x)dx = lim j-- 

 j ^ 



n 



That is, the definite integral of a function in a given interval 

 divided by the magnitude of the interval represents the arith- 

 metical mean of all the values of the function taken at equidistant 

 values of the variable throughout the interval, when the number 

 of values taken is increased indefinitely. 



From the definition it is evident that if/ (x) has the same sign 



rb 



throughout the interval ab, I f(x) dx has the sign of (6 a)/(X), 



J a 



and if there is in ab a finite interval cd in which f(x) is not zero, 



rd 

 then I f(x) dx is not zero. 



J c 



In particular, if the function is continuous in a whole interval 

 ab, and the integral between every two values of x in the interval 

 is zero, the function must be zero everywhere within the interval. 

 If therefore two continuous functions give in every interval ab the 

 same value of the integral, they must be equal everywhere in the 

 interval. 



Suppose that the continuous function f(x) has in the interval 

 ab a greatest value M and a least value m, the integral will have a 

 value lying between M(b a) and m(b a) and we may write, 



J a 



where M > A >m. 



Since f(x) is continuous, it will take the value A for at least 

 one value f of x between a and b, so that we may write 



(3) f/(*)e^ =/()(& -a), <<&. 



J a 



The above formula may be generalized. We have always 



(4) I (*/(*)* S (*<!/() 



I J a J a 



