In both baseball and mathematics education, the conventional wisdom is to avoid errors at all costs. That advice might be on target in baseball, but in mathematics, it is not always the best strategy. Sometimes an analysis of errors provides much deeper insights into mathematical ideas and, rather than something to eschew, certain types of errors can be extremely valuable. Numerical analysis has some lovely ideas that can enrich courses such as calculus while exposing students early to a deeper appreciation of numerical methods. In this article, the author focuses on ideas that can be incorporated into a freshman calculus course or used as a presentation at a mathematics club in conjunction with a discussion of numerical integration. The goal is to highlight the kinds of important mathematical insights that can be gained by examining the errors in numerical processes. In particular, the author looks at the issue of approximating a definite integral and seeks to obtain better approximations--that is, methods that are more accurate for a fixed number of subdivisions of an interval, so that they converge more rapidly. (Contains 7 tables and 4 figures.)