It is interesting to note that Simpson's Rule is the best polynomial approximation method for definite integrals in the general case. It might be intuitive to assume that higher-degree polynomial approximations to an integral would yield stricter error bounds, but this is not the case. I can personally vouch for this fact, as I was forced in high school calculus through the painful, ink-intensive process of deriving a general cubic approximation formula and its (more lax) error bound solely to stress the point that it just doesn't get any better than Simpson's rule.

To clarify: the polynomial approximations in question are nth-degree fits to (n+1)-point intervals, not general polynomials. The idea is that you take n+1 points evenly spaced out on the curve and fit a degree n polynomial to exactly those points. The whole point of this construction is that it is entirely algebraic. While a high-order Taylor approximation may often yeild better results, calculus is required to construct the approximating polynomial in the first place.