Though we will see in Chapter V that the result of IV.G can be generalized even further, we state here our main result in a slightly restricted form. We describe this theorem as "fundamental" because of the general and deep nature of its conclusion. Recall that we discussed at the beginning of this chapter that given a positive continuous function representing the velocity of a moving object, it is always possible determine the position of the object during its trip, i.e., a position function depending on time that changes at the specified velocity. The theorem gives a geometric version of this trip story. 

Theorem IV.4 The Fundamental Theorem of Calculus For Differential Equations (first draft).
[See Figure IV.H.i.]
If P is a positive continuous function on [A,B], then there is a function F with F'(t) = P(t) for all t where A<t<B.
In fact, F can be defined at x = t to be the area of the region enclosed by the X axis, the line X=A, the line X=t, and the graph of the function P, i.e., Y = P(X)^{[1]}Figure IV.H.i Comment: At this stage the details of the proof of Theorem IV.4 are not essential. The proof, given in IV.H Appendix, follows the general outline of the argument of Example IV.G.3. In our discussion so far we have used the notion of area of a planar region in an informal and intuitive sense. In Chapter V we will generalize our experience with area and Euler's method. This will give a more firm foundation to our use of area as an interpretation. A more general formulation will also allow other interpretations to take advantage of the mathematical results expressed in the Fundamental Theorems of Calculus as presented in Theorems IV.3 and IV.4.
Comments: 1. The approach to area problems discussed in Theorem
IV.3 and that used in the first solution to this example are not unrelated.
In the end each approach used the difference of the values of a solution
to the differential equation to determine the area. We will discuss these
two results again in Chapter V, but for now we note that one method used
approximations to make the connection between area and solving differential
equations while the other ties them together with the geometry of area
and the definition of the derivative.
2. Existence and Uniqueness. Theorem IV.4 states that when P
is a positive continuous function then the differential equation S'(t)
= P(t) has a solution. This result is described as an "existence"
result since it declares that a solution to the differential equation does
exist. After the work with tangent fields, Euler's method, and the heuristic
arguments about the trip, the result should not come as a surprise. It
and its generalizations are important because they allow us discuss solutions
even when they have no simple description as elementary functions. Theorem
4.3 has a quite different message, building on the ability to solve the
differential equation S'(t) = P(t). It declares that solving
the differential equation and computing the net change S(b) S(a)
can be interpreted in other contexts, like area, where measurement can
be estimated in a systematic way using sums of the form P(x_{0})^{.}
h + P(x_{1})^{.} h + P(x_{2})^{.}
h +...+ P(x_{ n1})^{.} h.
We can rephrase this result with a focus on the value S(b).
It says that value of a solution to the differential equation at b
is completely determined by the value at a, S(a) and function
P. In other words a solution to the differential equation S'(t)=P(t)
is determined uniquely by the equation and an initial or bounday condition.
It is in this way that Theorem 4.3 is considered a "uniqueness" result.
For the functions in problems 7 through 12, use the Fundamental Theorem of Calculus to find the area of the region enclosed by the Xaxis, the lines X = 1, X = 3 and the graph of the function.