Since ancient Greek times, mathematicians have attempted to compute areas and volumes as limits of approximations. The origins of the definite integral can be seen in Proposition 1 of Archimedes' Measurement of the Circle [16, pp. 91--93,]. In his proof, Archimedes computes the area of a circle from polygonal approximations using a clever double reductio ad absurdum argument combined with the ``method of exhaustion''.
The next major advance is found in a text of Cavalieri's [21, pp. 214--219,] illustrating his powerful ``method of indivisibles'' for computing the definite integral of simple polynomials. Cavalieri's book  was a very influential seventeenth century calculus text. While his method lacked rigor in part due to his cavalier attitude toward the infinite, he nevertheless succeeded in correctly computing many definite integrals.
Shortly thereafter, discovery of the inverse relationship between differentiation and integration transformed the definite integral into the most powerful computational tool in the mathematics and science of the time. Leibniz, in 1693, was the first to give a ``proof'' of the Fundamental Theorem of Calculus [21, pp. 282--284,], an intuitive geometric argument based on infinitesimals.
These ideas matured greatly in Cauchy's definition of the integral as a limit of sums in his series of calculus textbooks [5, vols. III and IV,] [11, pp. 566--571,], published in 1821--1823, which include his proofs of the most important theorems about the integral. Cauchy's methods are significant for two reasons: his departure from the traditional use of geometry to treat the definite integral, and his effective use of the developing concept of limit. By replacing a geometric definition by the power of algebra and the limit concept, Cauchy dispensed with the use of infinitesimals, and thus made more rigorous proofs of the basic theorems possible for the first time. Subsequently, Cauchy's work was put on a firm foundation by Weierstrass and his students, and generalized to apply to larger classes of functions via the Lebesgue integral.