The definite integral is one of the major
concepts of calculus; its relation to solving differential equations is
illuminated by the two forms of the Fundamental Theorem of Calculus
which are often visualized with the supporting justifications using the
area interpretation of the definite integral.

See Sensible Calculus:V.C The Fundamental Theorem of Calculus.(draft)

Nowhere in the standard treatments of the two forms of the Fundamental Theorem do mapping diagrams appear.

**Theorem CIS.FTC.I. Fundamental Theorem of Calculus I. (Derivative Form):**

Suppose $P$ is a continuous function on an interval, $I$, with $A$
any number in that interval. For any number $t\in I$, define the
function $S$ by $S(t)=\int_A^t P(x)dx$. Then for any $c \in I$,
$S′(c)=P(c)$.

**Theorem**** CIS.FTC.I**I. Fundamental Theorem of Calculus II. (Evaluation Form):

Suppose $P$ is a continuous function on an interval, $I$, with $a$ and
$b$ any numbers in $I$, and $F$ is an antiderivative (indefinite
integral or primitive function) for P$,$ i.e., $F′(t)=P(t)$ for all
$t\in I$. Then $\int_a^b P(x)dx=F(b)−F(a)$.

An important property of the definite
integral which is strongly related to these two forms of the Fundamental
Theorem and used in the proof of the derivative form is

**Theorem ****CIS.MVTI.**** The Mean Value Theorem for the Definite Integral.**

Suppose $P$ is a continuous function on $[a,b]$. Then for some $c\in (a,b)$, $P(c)[b−a]=\int_a^b P(x)dx$ or $P(c)=\frac{\int_a^b P(x)dx}{b−a}.$

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. It can also provide motivation and a heuristic proof of the evaluation form.

Before continuing further with this section it would be helpful to review the materials in
**CIS.EM.TMD**** on Euler's method.**

**CIS.VMVTI. Visualization of the Mean Value Theorem for Integrals**

**CIS.VFTC. Visualization of the Fundamental Theorem of Calculus
**