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Boole's Rule

Let the values of a function f(x) be tabulated at points x_i equally spaced by h=x_(i+1)-x_i, so f_1=f(x_1), f_2=f(x_2), ..., f_5=f(x_5). Then Boole's rule approximating the integral of f(x) is given by the Newton-Cotes-like formula

int_(x_1)^(x_5)f(x)dx=2/(45)h(7f_1+32f_2+12f_3+32f_4+7f_5)-8/(945)h^7f^((6))(xi).

This formula is frequently and mistakenly known as Bode's rule (Abramowitz and Stegun 1972, p. 886) as a result of a typo in an early reference, but is actually due to Boole (Boole and Moulton 1960).

See also

Hardy's Rule, Newton-Cotes Formulas, Simpson's 3/8 Rule, Simpson's Rule, Trapezoidal Rule, Weddle's Rule

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Boole, G. and Moulton, J. F. A Treatise on the Calculus of Finite Differences, 2nd rev. ed. New York: Dover, 1960.

Referenced on Wolfram|Alpha

Boole's Rule

Cite this as:

Weisstein, Eric W. "Boole's Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoolesRule.html

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