Common name | Short form | Exact for | |
0 | Midpoint rule | ||
1 | Trapezoid rule | ||
2 | Simpson's rule | ||
3 | ? | ? | |
4 | Bode's rule | ? | |
... | ... | ... | ... |
These are the actual Newton-Cotes formulas. The :s on the left signify on which monomial bases the solution is exact, e.g. a solution that is exact for will be exact on the monomials in the set . Note that the formulas' exactness increase in odd steps, so using the trapezoid rule for integrating is actually no better than using the midpoint rule, and just a waste of time. In the same manner, Simpson's rule will be exact for monomials in which will suffice for most functions found in practice.
If you also let the distance between each sample point vary, you obtain another group of quadrature formulas, the best of which is know as Gaussian quadrature.