While the primary audience for this book is an advanced undergraduate mathematics student, the contents will appeal to any mathematician who has wondered how the integrals introduced in elementary calculus and in real analysis courses fit together. By the time a young mathematician has completed the first year of graduate school, she will have encountered three versions of the integral: Riemann, introduced in elementary calculus; Darboux, studied in a first real analysis course and often still called a Riemann integral; and Lebesgue, developed in an advanced analysis course. Most often, these integrals are studied in isolation and with very little connection or comparison made between the different definitions. This book provides a comparative study of four approaches to integration over an interval [a,b]: Riemann, Darboux, Lebesgue, and gauge.

In addition to serving as a reference, this book can serve as a text for a second course in real analysis. Indeed, this manuscript is written with such users particularly in mind. The prerequisite first course should include the standard topics of supremum, infimum, compactness, the mean value theorem, and sequences of functions. The reader should also be familiar with using the formal “-ı definitions of limit and continuity in proofs. A series of appendices containing statements of the most relevant definitions and results from a first real analysis course is provided for readers who have encountered the requisite ideas but may need to refresh their memories. In addition, readers may find the notational index found at the beginning of the index helpful.

While the most celebrated milestone in the development of calculus comes from the late 17th century work of Newton and Leibniz, questions and ideas that lie at the heart of integral calculus were introduced by Eudoxus (4th century BCE) and Archimedes (3rd century BCE). The ideas of the differential and integral calculus (brought together by Newton and Leibniz) were powerful forces in the advancement of science. But cracks in the foundations of the subject (identified early on by Bishop Berkeley) became increasingly apparent toward the end of the 18th century. By the end of the 19th century, Cauchy, Riemann, and Darboux had addressed these foundational issues and had provided solid foundations for the integral calculus.