Editing Newton–Cotes formulas (section)

== Closed Newton–Cotes formulas ==
This table lists some of the Newton–Cotes formulas of the closed type. For <math>0 \le i \le n</math>, let <math>x_i = a + ih</math> where <math>h = \frac{b - a}{n}</math>, and <math>f_i = f(x_i)</math>.

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{|class="wikitable" style="margin:1em auto 1em auto; text-align: center;"
|+'''Closed Newton–Cotes Formulas'''
!{{mvar|n}} !!Step size {{mvar|h}} !!Common name !!Formula !!Error term
|-
|1 ||<math>b - a</math> ||[[Trapezoidal rule]] ||<math>\frac{1}{2} h(f_0 + f_1)</math> ||<math>-\frac{1}{12}h^3f^{(2)}(\xi)</math>
|-
|2 ||<math>\frac{b - a}{2}</math> ||[[Simpson's rule]] ||<math>\frac{1}{3} h(f_0 + 4f_1 + f_2)</math> ||<math>-\frac{1}{90} h^5f^{(4)}(\xi)</math>
|-
|3 ||<math>\frac{b - a}{3}</math> ||[[Simpson's rule#Simpson's 3/8 rule|Simpson's 3/8 rule]] ||<math>\frac{3}{8} h(f_0 + 3f_1 + 3f_2 + f_3)</math> ||<math>-\frac{3}{80} h^5f^{(4)}(\xi)</math>
|-
|4 ||<math>\frac{b - a}{4}</math> ||[[Boole's rule]] ||<math>\frac{2}{45} h(7f_0 + 32f_1 + 12f_2 + 32f_3 + 7f_4)</math> ||<math>-\frac{8}{945} h^7f^{(6)}(\xi)</math>
|}

Boole's rule is sometimes mistakenly called Bode's rule, as a result of the propagation of a typographical error in [[Abramowitz and Stegun]], an early reference book.<ref>[http://mathworld.wolfram.com/BoolesRule.html Booles Rule at Wolfram Mathworld, with typo in year "1960" (instead of "1860")]</ref>

The exponent of the step size ''h'' in the error term gives the rate at which the approximation error decreases. The order of the derivative of ''f'' in the error term gives the lowest degree of a polynomial which can no longer be integrated exactly (i.e. with error equal to zero) with this rule. The number <math>\xi</math> must be taken from the interval {{open-open|''a'',''b''}}, therefore, the error bound is equal to the error term when <math>f(\xi) = \max(f(x)), a<x<b</math>.