This book is an annotated catalog or encyclopedia of techniques in numerical integration, with the annotations giving particular attention to the types of function for which each method is best suited. It deals with integration only in the sense of evaluating integrals, and does not discuss numerical solution of differential equations. It concentrates on one-dimensional integrals, with one chapter on multiple integrals, and the treatment of Monte Carlo (sampling) methods is relatively weak. The chapter on error analysis is especially good, particularly since practitioners tend to take results on faith without any error analysis at all.

Like all good numerical analysis books, this one emphasizes that thinking should precede calculating: “Whenever possible, a problem should be analyzed and put into a proper form before it is run on a computer.’ (p. 5) Scattered throughout the book are examples of transformations that make an integral more tractable numerically. The book emphasizes the need to characterize the function before picking an integration method. The characterizations include not only an analysis of the singularities, but also broad behavioral characteristics such as whether the function is periodic, rapidly oscillatory, or has some special property such as satisfying a differential equation.

The book is an unaltered reprint of the 1984 second edition, but is still very up-to-date. For example, it covers all the integration methods used in *Mathematica* except for Duffy’s coordinates, and it is much more thorough (in its special subject of integration) than most numerical analysis texts. This is partly because it omits nearly all the proofs; the reader is referred to texts or original papers that contain the proof. The bibliography is enormous.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.