of Definite Integrals. 433 



if we now integrate each term between the given limits, and 

 restore for (p (log. a) its value f{a), we shall get the above 

 series. 



In the next theorem the limits of the integral are supposed 

 to be a, a + 2h, which may represent any whatever; the series 

 which expresses the value of the definite integral of any function 

 is not only remarkable for the simplicity of its form, but likewise 

 because it gives the law of the errors made by taking any number 

 of its first terms for the definite integral. 



Theorem III. Let cp„{a) denote the error made by taking the 

 first w— 1 terms of the following series for the true value of the 

 definite integral of ^(x); then shall 



^ r >. / , ^^g 3 da 3 da 



h d.(j)Ja) h d.(pja) 

 5 ' da 5 ' da 



- &c. 



Thus if ^(1) be the ordinate of a curve, D.ip{x) is the area con- 

 tained between the curve, the two extreme ordinates, and the 

 axis of X, the first term represents only the inscribed parallelogram, 

 the two fir.st represent the trapezoid formed by the chord, the 

 extreme ordinates, and the axis of.!' ; the three first terms represent 

 the parabolic segment, which has its extreme points common with 

 the curve, which was the approximation used by Simpson, and 

 the remaining terms give always the nearest approximations (when 

 /' is small) that may be made to the true area. For if we suppose 

 ,f{x) to be the indefinite integral, and u to be the true value of 

 the definite integral, we rau.st have 



/(« + 2h) -/(a) = u, 



and if we expand the first term by Taylor's theorem, we shall 

 net a linear e<piation of infinite dimensions, which being resolved 



3k2 



