The definite integral is one of the major concepts of calculus; its study often begins in the first semester of a calculus course and can continue through much of the study of mathematics and its application in almost every science.
See Sensible Calculus: V.A The Definite Integral and V.E Riemann Sums and Definition of the Definite Integral.
In many calculus courses, the definite integral is motivated from the problem of finding the area of a region in the plane. The definition is presented as a limit of sums related to estimating areas for the motivating problem. Basic properties of the definite integral are visualized with the area motivation, leading eventually to two fundamental theorems: a derivative form related to the area visualization, and an evaluation form justified with the derivative form and a connection to differential equations.

Nowhere in the standard treatments of the definite integral do mapping diagrams appear: not in the definition nor in the two forms of the fundamental theorem.

An interpretation of the definite integral as a limit of sums in solving an initial value differential equations problem with Euler's method provides an alternative visual motivation and a  basis for understanding the definite integral as a significant part of the study of rates in many contexts.
Before continuing with this section it would be helpful to review the materials in CIS.EM.TMD on Euler's method.