In finding exactly and estimating functions to solve differential equations we have seen another concept taking shape to understand change. In somewhat abstract terms, we want to find the net change in the value of a controlled (dependent) variable for a given interval of the controlling (independent) variable. The consolidation (integration) of information about a controlled variable's rate of change, `{dy}/{dx}=f '(x)`, with respect to a single controlling variable, x, for the interval [a,b], leads to the recovery of information about the net change in the controlled variable's values, `Delta y = f (b) – f(a)`. This context is common enough to articulate it more carefully and distinguish it from those already developed in solving differential equations.
In our study we found that by accumulating estimates of change in a controlled variable (dy ` ~~ Delta`y) over small intervals of change in the controlling variable (dx = `Delta`x) we could make reasonable estimates of the net change in the controlled variable over a longer interval ( [a,b]). In this chapter we will define and begin to study the concept called the definite integral that captures this procedure of accumulating estimates and the number estimated by this process.
The sums used in Euler's method (Chapter IV.***) to estimate the solution
to a differential equation of the form f '(x) = P(x)
where P was a continuous function had at least two interpretations, one
related to motion and one related to areas.
In the motion interpretation these sums estimated the net change in a moving object's position, S, during the time interval [a,b], i.e., S(b)  S(a), based on the model assumption that its velocity at time t was given by v(t) = S'(t) = P(t). See Figure 1. 

In the geometric interpretation when P(x) > 0 the sums estimated the area of the region in the coordinate plane enclosed by the X axis, the lines X = a and X = b, and the graph of the function P, i.e., the graph of the equation Y = P(x). See Figure 2. The fact that these sums appear to approach a single number as their limit and the importance of these sums in a variety of interpretations has led to the articulation of the mathematical concepts involved. 

Notation: Suppose that P is a function defined on the interval [a,b] and N is a positive integer. We will use let dx = `Delta x =h =(ba)/N`. With this established we continue by denoting a by x_{ 0}, so a = x_{ 0}, and then we let x_{ 1} = a + `Delta`x,
x_{ 2}= x_{ 1} + `Delta`x = a + 2 `Delta`x ,
x_{ 3}= x_{ 2} + `Delta`x = a + 3 `Delta`x .
We continue this notation simply by letting x_{ k}
= a + k `Delta`x. Note
that this means that x_{ N} = a + N `Delta`x
= b.
Since the numbers x_{ k }arose in the context
of Euler's method for solving a differential equation,
we describe
the set {`a = x_0 , x_1 , x_2 , ... , x_N =b`} as an Eulerian partition
of the interval [a,b] with norm or mesh dx = `Delta x
=h =(ba)/N`.
Notes:
1. We have assumed that a<b, so that dx = `Delta`x = h > 0 and thus `a = x_0 < x_1 < x_2 < ... < x_N =b`.
2. As the size of N increases, the size of `Delta`x = h approaches 0. Symbolically, as N ` > oo, Delta x = h > 0`.
3. If we had chosen a and b without the presumption that a<b, all
the notation would still make sense, though if b<a then `Delta`x<0
and
`b = x_N < x_{N1} < ... < x_1 < x_0 =a`.
As we have said before, sums were important quantities in estimating the net change in position and the area in our previous work, so we establish a shorthand notation for these important numbers as well by letting
We'll refer to these sums as the N^{th} Eulerian sum for
the function P over the interval [a,b] or the Eulerian
sum for short.
More Notes: 1. The Eulerian partition of [a,b]
breaks the interval into N distinct segments (sub intervals). In the Eulerian
sums we evaluate the function P at the left hand endpoint of each of these
intervals. The left hand endpoint of the first segment is x_{
0}, the left hand endpoint of the second segment is x_{
1}, the left hand endpoint of the third segment is x_{
2}, and the pattern continues with the left hand endpoint of the
k^{th} segment being x_{k  1}, so that last
summand of S(P,n) evaluates P at the lefthand endpoint of the (last) N^{th}
segment, namely x_{N  1}.
2. The Democracy Principle : "What's good for one is good for all."
Essentially the sum is the accumulation of the results of the same
computational process applied to each subinterval: Evaluate P at the left
hand endpoint of the segment, then multiply that result by `Delta`x. This is one of the key principles for all of mathematics! 

Historical Comment: The precise definition of the definite integral has had a long history and as with many mathematical concepts, with its continued refinement it has gained in abstraction and power. It was Augustin Cauchy who is generally credited with bringing some precision to these concepts in the early 19th century, but the development continued to its generally acknowledged first completion in the mid 19th century by the German mathematician Georg Friedreich Bernhard Riemann (1826 1866). Only in the beginning of the 20th century did the definite integral reach its full generality with the work of the French mathematician Henri Lebesgue (18751941) on what is now described as the theory of measure. The notion of the definite integral has continued to grow further even in the 20th century, demonstrating the vitality of the concept in an ever expanding world of mathematical studies. 
which results from recognizing the common factor of `Delta`x in equation *.
4. Although the notation ignores the relation between the number N and
the other subscripts used in denoting the numbers in the partition of [a,b],
don't be fooled into thinking that the partitioning numbers are the same
for different N. For example, for the interval [0,5] the Euler partition
with N = 5 is {0,1,2,3,4,5} so with N= 5 it turns out that x_{k}
= k. But with N=4, the Euler partition is {`0, 5/4, 5/2, 15/4, 5`} or x_{k}
= `(5k)/4`.
The notation ignores the different values of N, but it is important
to recognize this, since it means that data used to compute with N= 5 will
be different from that used to compute with N=4.
If we wish to use data computed when N=5 in the computation of S(P,N)
for other values of N, 5 must be a factor of N.
Here is another place where bisection or decimation can come in handy
as a technique for computing that does not discard old information.
5. Our treatment here has emphasized the motion and area
interpretations
of the definite integral, but we should not forget that the
differential
equation tangent field also can be used to visualize these sums. In
particular
we can interpret S(P,N) using a tangent field for the differential
equation `(dy)/(dx) = P(x)` and estimating an integral curve for this
field
with line segments. See Figure 3. We can consider this estimate as an approximation for the graph of a
position function s that satisfies the differential equation. Now
locate the points on the estimating curve with first coordinates a
and b. Assume these points have coordinates (a,c) and (b,d). The
interpretation of the sum as the net change in position is now seen as
the accumulation of the step by step vertical changes made by this estimating
curve over the interval [a,b]. Thus we see as well that `S(P,N) ~~ d  c`
and represents an estimate of the net change in the second coordinates for an integral curve fitting this field. 

Example V.A.1. In this example we consider some specifics in
computing the Euler sums.
Suppose P(x) = 3 x^{ 2} and a = 1 while
b = 5 with N = 4. Then `Delta x
= (5 1)/4 = 1` and so x_{ 0} = 1 , x_{ 1}
= 2, x_{ 2} = 3 , x_{ 3} = 4, and x_{
4} = 5.
Thus, S(P,4) = P(x_{ 0}) `Delta`x + P(x_{ 1}) `Delta`x + P(x_{ 2})`Delta`x + P(x_{ 3})`Delta`x
= 3 + 12 + 27 + 48 = 90.
Using N = 5, gives `Delta x = (5 1)/5 = 4/5 `, so x_{ 0} = 1 , x_{ 1} = `9/5`, x_{ 2} = `(13)/5` , x_{ 3} = `(17)/5`, x_{ 4} = `(21)/5`, and x_{ 5} = 5.
Finally, S(P,5) = [P(x_{ 0}) + P(x_{ 1})
+ P(x_{ 2}) + P(x_{ 3}) + P(x_{
4})] `Delta x = [3 + 3 (81)/25 + 3 (169)/25
+ 3((17)/5)^2 +3((21)/5)^2 +75](4/5) ~~ 98.88`.


Of course much larger values of n would be even more difficult without
the aid of a calculating procedure or better algebra.
Here is a table showing
the results of further computation of S(P,N) in a decimating scheme.





















Theorem V.2 (The Fundamental Theorem of Calculus  Evaluation Form)
Evidence for Theorems V.1 and V.2:
Motion Interpretation. Consider P as a velocity function which changes continuously for the time interval [a,b]. We can consider the sums S(P,N) as the result of selecting the velocity values at a finite number of points in the time interval, using these velocities to estimate the change in position for short time intervals, and then accumulating these estimates for an estimate of the net change in position for an object moving with velocity P for the time interval [a,b]. Since the velocity is a continuous function we can perform a thought experiment setting controls for an object to move with the given velocities. When the object in this experiment moves we can keep track of its position with the function F. Its net change in position for the time interval [a,b] will be the number F(b)F(a). This is the number that the sums S(P,N) approach when N is very large.  Cost Interpretation. Consider P as a marginal cost function which changes continuously for the production level interval [a,b]. We can consider the sums S(P,N) as the result of selecting the marginal cost values at a finite number of points in the proction level interval, using these marginal costs to estimate the change in cost for short production level intervals, and then accumulating these estimates for an estimate of the net change in costs for production for increasing the production over the interval [a,b]. Since the marginal cost is a continuous function we can perform a thought experiment setting controls for production to change with the given marginal costs. When production in this experiment changes we can keep track of its total costs with the function F. Its net change in costs for the production interval [a,b] will be the number F(b)F(a). This is the number that the sums S(P,N) approach when N is very large. 


A Little More Notation: As we have seen in chapter IV, the evaluation of the difference F(b)  F(a) is both useful and common. For this reason the older notation for evaluation of a variable has been extended to denote this difference by `F(x) _{x=a}^{x=b} = F(x)  _a^b = F(b)  F(a)`. So for example in the last example the work might have omitted the naming of the function F by displaying F(x) instead as follows:
Frequently Asked Questions:
1. Can we always find an antiderivative for P as an elementary function?
This would be quite nice, matching the niceness of the derivative calculus where we have rules that allow us to find the derivative of any elementary function. Unfortunately this is not the case. Functions as simple as `sin(x^2)` and ` e^{x^2}` do not have antiderivatives that can be expressed as an elementary function. [The proof of this fact is not easy.] As a result there is no easy way to find `int_0^1 sin(x^2) dx` or `int_0^1 e^{x^2} dx` by applying Theorem V.2.
2. Is there a geometric interpretation for the definite integral when
the integrand Q is negative on the interval [a,b]?
When considering continuous functions over a compact interval, several
alternative definitions can be used to estimate the same number that
we have defined using the left endpoint Euler sums.
Here a some of those
alternatives with figures (to be added still!) illustrating their meaning in terms of both the
motion and the area interpretations of the estimates.
Euler: Let `Delta x = (b  a)/n`
; `x_ k = a + k Delta x`, `k = 0, 1,... n.`
`int_a^b P(x)d x = lim _{n > oo} S_n (P,n)`


Extended Euler: Choose numbers `w_k` so that
`x_{k1}<= w_k <= x_k ;
k = 1, ..., n` .
Let W = { w_{ 1}, w_{ 2}, ... ,w_{ n1}, w_{ n }}. Now we let `S _n (P,n,W) = sum _{k=1}^{k = n} P(w _k) Delta x`. Then we define the definite integral by `int_a^b P(x)d x = lim _{n > oo} S_n (P,n,W)` 
Let W = { w_{ 1}, w_{ 2}, ... ,w_{
n1}, w_{ n }}and let `Delta`x = mesh of the partition V = max {`Delta`x_{
k} , k = 1 to n}
Now we let `S _n
(P,W) = sum _{k=1}^{k = n}
P(w _k) Delta x_k`. Then we define the definite integral by
`int_a^b P(x)d x = lim _{Delta x > 0} S_n (P,W)`
`int_a^b P(x) dx = ` I, a unique number with the property that for any partition, `L_n (P,n) <=` I ` <= U_n (P,n)`.
Using an Antiderivative: Suppose G'(x) = P(x) for all x in [a,b]. Then we define the definite integral by `int_a^b P(x) dx = G(b)  G(a)`.
CAVEAT!
This makes a definition of the Fundamental Theorem so the Fundamental
Theorem must be restated in terms of connecting the definite integral
to some kind of sums!
`S(P,N) = 3 (1)^2 4/N + 3 (1+4/N)^2 4/N + 3 (1+8/N)^2 4/N + ... +3 (1+ ((N1)4)/N)^2 4/N`
`... = 3 [ N + 4N(N1)/N + 16 ((N1)N(2N1))/(6N^2) ] 4/N `
`... = 124  (144)/N +(32)/(N^2)` .