Subsection CIS.DI The Definite Integral

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.

DEF.CIS.DI. Definition of the Definite Integral.

CIS.VDI. Visualization of the Definite Integral

CIS.PDI. Properties of the Definite Integral