Over three decades ago, Stroud published his encyclopedic work on multiple numerical integration, Approximate Calculation of Multiple Integrals [Str71]. In his book, Stroud presented a rather complete summary of the theoretical and practical aspects of multiple numerical integration, a comprehensive bibliography and a listing of almost all multiple integration or cubature rules for a variety of regions. This book has proved very useful to all workers in the field and very few papers concerned with multiple numerical integration omit it from their list of references.
About 8 years ago, Cools and Rabinowitz [CR93] published a compilation of all so-called monomial cubature rules which appeared since the publication of [Str71] plus some cubature rules which appeared earlier but were not included in [Str71] for some reason. This compilation includes references to cubature rules but not the actual points and weights of these cubature rules. The word `all' above must be qualified in several ways. First, they restricted themselves to four regions so that several regions treated in [Str71] are not included in their compilation such as the surface of the sphere, the spherical shell, the hexagon, the octahedron, etc. Second, they have ignored the Russian and other non-western literature except when it has appeared in translation. Third, they have not included every cubature rule, omitting cubature rules which did satisfy certain reasonable criteria which given below. Finally, they surely overlooked some cubature rules which have appeared in the accessible literature and, a fortiori, cubature rules which have appeared in theses, technical reports, research papers, etc.
The purpose of these pages is to continue the work by Stroud [Str71] and Cools and Rabinowitz [CR93]. These pages give an overview of all known cubature rules for the four regions considered in Cools and Rabinowitz [CR93]. Besides references to the literature, the points and weights of the cubature rules are given and, at least in some cases, some additional information.
In the next section, we give the necessary background and description of the tables including some guidelines for deciding which cubature rules were included and which sources are given for a cubature rule when it has several sources.