ABSTRACT
Integration is one of the most common of scientific computations. Sometimes a symbolic mathematical system or sheer ingenuity allows us to derive a closedform solution (i.e., an algebraic equation) for an integral. However, the integrals of many simple functions do not have simple solutions, and consequently much effort has been devoted to performing numerical definite integration robustly and quickly. In this chapter we shall take a brief tour through the great variety of integration techniques, and we shall explore the interpolation mechanisms we developed earlier as possible bases for good numerical integration routines. In particular, we shall discuss Riemann sums, trapezoid rule, Simpson’s rule (and other approaches based on Lagrange interpolation), and smooth cubic interpolation. These correspond to polynomial interpolation of degrees zero to three. We then show how to extend the techniques to compound or piecewise integration. Several case studies will be considered, but an error analysis will be deferred to the next chapter. Finally, we shall briefly consider the computation of integrals using a completely different approach: flipping coins. These are called Monte Carlo methods. We used the phrase computational integration for the chapter title rather than numerical integration or quadrature so as to include the symbolic computation of integrals. While it is certain that traditional numerical integration techniques will remain unsurpassed for efficiency and robustness, the ease with which symbolic systems can be used to reason about and to explore properties of integrals makes them important partners in the effort to solve integration problems.
