Integration, a genetic introduction

The genetic method of O.Toeplitz is a structuring of conventional instructional material using historical insights and historical problems, not for a history course, but for the better understanding of modern mathematics by students today. The paper which follows describes an introduction to integration suitable for a first course in analysis, in the style of Toeplitz. After two examples of the Greek method of exhaustion (intended for lecturers rather than students), historical developments from the 17th and 19th centuries pinpoint the conceptual steps which a student must take in a conventional first course in analysis up to the Riemann integral.