Trapezoidal, simpsons rule, and higher order approximations

[okkvlt]

hi. i was able to prove the trapezoidal rule and simpsons rule. (basically i used matrices to determine the coefficients m and b for mx+b when proving the trapezoidal rule and a,b,c for ax^2+bx+c such that the points coincide, then i integrated the approximating polynomial) the amount of number-crunching and expanding products of linear terms was probably the most work I've ever done, but i was amazed to see terms cancel out to yeild vastly simplified formulas. but i was wondering, suppose i approximated each step with suppose, a cubic equation. i know how to do this, but after setting up the 4x4 matrix i quickly realized that this would be extremely mundane due to all the algebraic manipulations. so before i try deriving a numerical method that uses cubics to approximate the curve pieces, i want to know whether it will simplify. will it simplify?

 

[arildno]

You can look upon Newton-Cotes formulae, and Gaussian quadrature to see how these numerical techniques have been developed:
http://mathworld.wolfram.com/Newton-CotesFormulas.html