Cauchy definition of definite integral

This note is about Cauchy’s (real-variable) approach to the definite integral as a limit of sums, historically preceding the modern formulation of the Riemann integral. (Not to be confused with the complex-analytic Cauchy integral formula.)

Origins (very brief)

Archimedes: method of exhaustion via upper/lower approximations of areas/volumes.
Kepler / Cavalieri / Wallis: infinitesimal-style decompositions and “indivisibles”.
Newton–Leibniz: integral as inverse operation to differentiation.
Cauchy: returns to area-style definition and makes limit language explicit.

Cauchy sums

Assume $f$ is continuous on $[x_{0}, X]$ and take points

x_{0} < x_{1} < \dots < x_{n - 1} < X .

Cauchy considers sums like

S = \sum_{i = 0}^{n - 1} f (x_{i}) (x_{i + 1} - x_{i}), (x_{n} := X)

and argues that as the subdivision gets finer (the maximum $max_{i} (x_{i + 1} - x_{i})$ tends to $0$ ) the value of $S$ approaches a limit depending only on $f, x_{0}, X$ .

He also notes one can choose sample points inside each subinterval, i.e.

S = \sum_{i = 0}^{n - 1} f (x_{i} + ϑ_{i} (x_{i + 1} - x_{i})) (x_{i + 1} - x_{i}), ϑ_{i} \in [0, 1],

which is essentially the tagged-sum viewpoint later used by Riemann.

Conceptual caveat (modern viewpoint)

Varying the partition does not naturally produce a sequence; it forms a directed family (a net) indexed by refinements. This is one place where the original 19th century discussions can look imprecise by modern standards.

Beyond continuity (Cauchy/Dirichlet/Lebesgue)

Cauchy also sketches extensions beyond continuous functions by splitting the interval into pieces where the function behaves well, and defining the integral by limits near singular/discontinuous points.

Dirichlet extends this idea to certain functions with infinitely many discontinuities but only finitely many accumulation points.

A key limitation (as discussed by Lebesgue) is that the Cauchy–Dirichlet approach essentially requires an antiderivative-type representation on intervals of continuity and restricts the discontinuity set (e.g. to be countable).