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The usual applications of the integral, viewed as an additive function over the subintervals of \([a,b]\), are easy to obtain. The justification for the additive property can be accomplished through the use of heuristics in the setting of known applications of the integral, such as arc length, area, volume, work, area between two curves, etc. In all these cases the additive property is obvious. The boundedness condition is evident and easily established in many cases, such as area or work. For instance, the work completed by a one dimensional variable force applied on a straight line is bounded below by the work done if we apply the minimum value of the force during the whole trajectory, and bounded above by the work done if we apply the maximum value of the force during the whole trajectory.

In other cases the required bounds are not so clear. Gillman [20, p. 17] stated that the axiomatic presentation of the integral had been done previously with varying degrees of thoroughness, and described among his own contributions to this approach that of providing an improved version of Property (B) of Definition 1, useful in setting up integrals where the desired bounds were not so obvious. In the setting of the present approach to the integral, all applications are obtained by following the same basic pattern. First, one identifies the quantity to be represented as an integral \(I_{a}^{b}\) and verifies that properties (A) and (B) are satisfied. This is accomplished using knowledge from the field of application or, sometimes, on heuristic grounds. As Gillman himself remarked, “From then on, the rest is mathematics” [20, p. 20]. The desired quantity is the unique integral satisfying conditions (A) and (B).

**Figure 2.** Estimation of Bounds I

In order to show Gillman's improvement on condition (B) of Definition 1 we look at an example where the basic pattern does not work. For instance, in Figure 2, we have two functions \(y=f(x)\) and \(y=g(x)\) defined over a portion of an interval \([a,b]\) such that \(f(x)\geq g(x)\geq 0\) over that interval. In determining the volume of the solid of revolution \(V_{u}^{v}\) around the \(x\) axis of the area bounded by both curves lying between \(x=u\) and \(x=v\), it is easy to obtain the following bounds: \[\pi\cdot\big[\min_{w\in [u,v]}f^{2}(w)-\max_{w\in [u,v]}g^{2}(w)\big](v-u)\leq V_{u}^{v}\quad\quad\quad\quad\quad\quad\notag\] \[\leq\pi\cdot\big[\max_{w\in [u,v]}f^{2}(w)-\min_{w\in[u,v]}g^{2}(w)\big](v-u).\quad\quad\quad(6)\] These bounds are not too useful to begin with, and in some cases, like the one depicted in Figure 3, expressions like \[\big[\min_{x\in [u,v]}f^{2}(x)-\max_{x\in [u,v]}g^{2}(x)\big]\] are negative.

**Figure 3.** Estimation of Bounds II

To circumvent this difficulty Gillman [20, p. 20] invented an ingenious limit bounding condition:

(B') **Gillman's limit bounding condition.** Let \(f:[a,b]\rightarrow\mathbb{R}\) be continuous and suppose \([u,v]\mapsto J_{u}^{v}(f)\) (\(a\leq u\,{\rm{<}}\,v\leq b\)) is a function such that for each \(x\in [a,b]\) and \(u\leq x\leq v\) (\(u\,{\rm{<}}\,v\)) there are real parameters \(\phi_{u,v}\) and \(\varphi_{u,v}\) depending on \(x\) such that \[\phi_{u,v} \cdot (v-u)\leq \,J_{u}^{v}(f)\leq\varphi_{u,v}\cdot (v-u),\] and \[\lim_{u,v\to x}\phi_{u,v}=\lim_{u,v\to x}\varphi_{u,v}=f(x).\] If \(x=a\) or \(x=b\), we interpret the condition accordingly so as to have one-sided limits.

We can easily see that the following theorem is true.

**Theorem 3. **If \(f\) has an integral on \([a,b]\) and \([u,v]\mapsto J_{u}^{v}(f)\) is a function defined on the subintervals of \([a,b]\) satisfying condition (A) of Definition** **1 and Gillman's limit bounding condition (B'), then \(J(f)\) is the (unique) integral for \(f\) on \([a,b]\).

** Outline of the proof.** By part (A) of Definition 1 and by condition (B'), it is easy to see that \[\frac{d}{dx}J_{a}^{x}=f(x).\] If \(x\mapsto I_{a}^{x}(f)\) is an integral for \(f\) on \([a,b]\), then it must differ by a constant from \(x\mapsto J_{a}^{x}(f)\). Since \(J_{a}^{a}(f)=I_{a}^{a}(f)=0\), then \([u,v]\mapsto I_{u}^{v}(f)\) and \([u,v]\mapsto J_{u}^{v}(f)\) must be the same integrals, which establishes the theorem.

Going back to our discussion of relation (6), we see that \[\lim_{u,v\to x}\big(\min_{w\in [u,v]}f^{2}(w)-\max_{w\in [u,v]}g^{2}(w)\big)=f^{2}(x)-g^{2}(x),\] and also that \[\lim_{u,v\to x}\big(\max_{w\in [u,v]}f^{2}(w)-\min_{w\in [u,v]}g^{2}(w)\big)=f^{2}(x)-g^{2}(x).\] It follows that \[V_{a}^{b}=\pi\int_{a}^{b}[f^{2}(x)-g^{2}(x)]\, dx\] gives the volume of revolution around the \(x\) axis. This is the well known “washer method” for the calculation of volumes of revolution. When the area depicted in Figure 2 is rotated about the \(y\) axis, we get the estimates \[\pi(u+v)\big(\min_{w\in [u,v]}f(w)-\max_{w\in [u,v]}g(w)\big)(v-u)\leq V_{u}^{v}\] \[\leq\pi(u+v)\big(\max_{w\in [u,v]}f(w)-\min_{w\in [u,v]}g(w)\big) (v-u).\] Using (B') again, we get \[V_{a}^{b}=2\pi\int_{a}^{b}x[f(x)-g(x)]\,dx.\] This is the expression for the so-called “shell method” for calculating volumes of revolution.

Gillman [20] presented other examples of heuristic arguments for circumventing the difficulties of applying the estimate (B) of Definition 1.

Omar A. Hernandez Rodriguez (University of Puerto Rico) and Jorge M. Lopez Fernandez (University of Puerto Rico), "Teaching the Fundamental Theorem of Calculus: A Historical Reflection - Standard Applications of the Integral," *Convergence* (January 2012), DOI:10.4169/loci003803