The definite integral has an interesting history. In this thesis we trace its development from the time of ancient Greece (500-200 B. C.) until the modern period. We place special emphasis on the work done in the nineteenth century and on the work of Lebesgue (1902). The thesis is divided into four parts arranged roughly chronologically. The first part traces the developments in the period from the fifth century B. C. until the eighteenth century A. D. Secondary sources were used in writing this history. The second part recounts the contributions of the nineteenth century. The original works of Cauchy, Dirichlet, Riemann, Darboux, and Stieltjes are examined, the third part is concerned with the development of measures in the latter part of the nineteenth century. This work leads to the Lebesgue integral. The final part is a brief survey of modern ideas. / Science, Faculty of / Mathematics, Department of / Graduate