Simpson's Rule Is Exact for Quintics

1. INTRODUCTION. Everyone who has taught the error estimate for Simpson's rule in a few freshman calculus courses knows that the chances are very good that someone who hasn't quite thought things through will look at the error estimate and then ask " Yes, but how can we find the error exactly? " The questioner is usually thunderstruck by the observation that if we knew how to do so in general, then we wouldn't speak of " Simpson's rule for approximating integrals, " but of " Simpson's rule for evaluating integrals. " Nevertheless, under certain circumstances we can find exact error. It is well known, for instance, that Simpson's rule error is zero whenever the integrand is a polynomial of degree three or less. What is less well known is the fact that we have adduced in our title: as we show in this article, we can easily find the exact error (see equation (5)) for Simpson's rule approximations to integrals of polynomial functions of degree four or five. We accomplish some other things as well, including an extension of the theorem generally known as the " First Mean Value Theorem for Integrals. " We begin by reviewing some background. We all learned in our elementary calculus courses that if K 2 is a bound for | f