  Differential lambda-calculus was first introduced by Thomas Ehrhard and
Laurent Regnier in 2003. Despite more than 15 years of history, little work has
been done on a differential calculus with integration. In this paper, we shall
propose a differential calculus with integration from programming point of
view. We show its good correspondence with mathematics, which is manifested by
how we construct these reduction rules and how we preserve important
mathematical theorems in our calculus. Moreover, we highlight applications of
the calculus in incremental computation, automatic differentiation, and
computation approximation.
