Mathematical models, based on ordinary, partial differential, integral, and integro-differential equations, are the only viable tool for studying our physical surroundings and their natural manifestations. Therefore, it is important for practitioners to strive to find solutions to these models, either analytic or numerical. This two-volume set gathers up-to-date research results that show how to set up a very important class of such tools, and how to use them in specific problems of science and engineering.The two volumes contain 65 chapters, which are an outgrowth of talks presented by experienced researchers from 24 countries on 5 continents at the Tenth International Conference on Integral Methods in Science and Engineering held in Santander, Spain, July 7-10, 2008. The chapters address a wide range of topics, from the construction of boundary integral methods to the application of integration-based analytic and numerical techniques to mathematical models arising in almost all aspects of today's technological world-in particular, integral equations, finite and boundary elements, conservation laws, and hybrid approaches.Volume 1: Analytic Aspects, contains the first 31 of these chapters and is followed by Volume 2: Computational Aspects. Volume 1 covers applications to a wide spectrum of subjects, ranging from deformable structures, to traffic flow, acoustic wave propagation, spectral computation, among others.Integral Methods in Science and Engineering, Volume 1 is a useful research guide for pure and applied mathematicians, physicists, biologists, and mechanical, civil, and electrical engineers, graduate students in these disciplines, and various other professionals who use integration as a fundamental technique in their research.