Let the values of a function 
be tabulated at points 
equally spaced by 
, so 
, 
, ..., 
. Then Woolhouse's formulas approximating the integral
of 
are given by the Newton-Cotes-like formulas
 |  | ![5[(223)/(3969)(f_1+f_(11))+(5875)/(18144)(f_2+f_(10))+(4625)/(10584)(f_4+f_8)+(41)/(112)f_5]](/images/equations/WoolhousesFormulas/Inline10.svg) |
(1)
|
 |  | ![14[7/(195)(f_1+f_(29))+(16807)/(66690)(f_3+f_(27))+(128)/(285)(f_8+f_(22))+(71)/(135)f_(15)].](/images/equations/WoolhousesFormulas/Inline13.svg) |
(2)
|
Woolhouse's Formulas
Explore with Wolfram|Alpha
References
King, A. E. "Approximate Integration. Note on Quadrature Formulae: Their Construction and Application to Actuarial Functions." Trans. Faculty of Actuaries 9, 218-231, 1923.Sheppard, W. F. "Some Quadrature-Formulæ." Proc. London Math. Soc. 32, 258-277, 1900.Whittaker, E. T. and Robinson, G. "Woolhouse's Formulae." The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 158, 1967.Woolhouse, W. S. B. "On Integration by Means of Selected Values of the Function." J. Inst. Act. 27, 122-155, 1888.Referenced on Wolfram|Alpha
Woolhouse's FormulasCite this as:
Weisstein, Eric W. "Woolhouse's Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WoolhousesFormulas.html