Next, let's use three terms in the weighted sum. The set of polynomials will also be extended to include the second power, so that the set becomes . Consequently, we have three integrals to require are exactly captured with the three terms in the weighted sum. This entails the following:
This represents three equations in three unknowns and can be solved by hand or using Mathematica. The weights are
Then, the Simpson's rule integration formula is:
Thus, if the interval size h is not very large, the Simpson's rule formula is another order of magnitude better in accuracy than the trapezoid rule, provided that the derivative in the error does not become too large too fast. However, when h is not small it is pretty clear that this formula is not likely to be of much use. We shall discuss how to remedy this situation below.