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**