ABSTRACT
The direct fit polynomial procedure presented in Section 6.2 requires a significant amount of effort in the evaluation of the polynomial coefficients. When the function to be integrated is known at equally spaced points, the Newton forward-difference polynomial presented in Section 4.6.2 can be fit to the discrete data with much less effort, thus significantly decreasing the amount of effort required. The resulting formulas are called Newton-Cotes formulas. Thus,
where P,,(x) is the Newton forward-difference polynomial, Eq. (4.88):
( ) -1' /11' s(s - I) /121' s(s - I)(s - 2) /131'P" X - JO + s YO + 2 JO + 6 JO s(s - I)(s - 2) .. · [s - (n - I)] "
where the interpolating parameter s is given by x -xo
s =-- ---+ x = Xo + shh
(6.13)
(6.14)
(6.15)
(6.16)
and the Error term is
Equation (6.13) requires that the approximating polynomial be an explicit function of x, whereas Eq. (6.14) is implicit in x. Either Eq. (6.14) must be made explicit in x by introducing Eq. (6.16) into Eq. (6.14), or the second integral in Eq. (6.13) must be transformed into an explicit function of s, so that Eq. (6.14) can be used directly. The first approach leads to a complicated result, so the second approach is taken. Thus,
1= I(x) dx e:' P,,(x) dx = h PIles) ds a a s(a)
where, from Eq. (6.15) dx=hds
(6.17)
(6.18) The limits of integration, x = a and x = b, are expressed in terms of the interpolating parameter s by choosing x = a as the base point of the polynomial, so that x = a corresponds to s = 0 and x = b corresponds to s = s. Introducing these results into Eq. (6.17) yields
1 = h J: PIl(xo +sh) ds (6.19) Each choice of the degree n of the interpolating polynomial yields a different
Newton-Cotes formula. Table 6.1 lists the more common formulas. Higher-order formulas have been developed [see Abramowitz and Stegun (1964)], but those presented in Table 6.1 are sufficient for most problems in engineering and science. The rectangle rule has
Formulas
poor accuracy, so it is not considered further. The other three rules are developed in this section.
