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Which bounded functions are integrable?

This is a difficult question to answer. One simple partial answer is that monotonic functions are integrable.

A function is monotonic on an interval if it is either increasing or decreasing on the interval. If f is a monotonic function on [a,b] then it is bounded and integrable. (And the result can be easily generalized to bounded piecewise monotonic functions [see Ross, p. 196]).

"Fortunately, most of the functions that occur in practice are monotonic or sums of monotonic functions, so the result of this miniature theory of integration is quite comprehensive." (Apostol, p. 76)

In the applet you can play with a simple case of monotonic functions: continuous monotonic functions, but the result is true in general. (By the way, another important family of integrable functions are all continuous functions on [a,b]).

In the applet it is possible to modify the curve and the ends of integration, to increase or to diminish the number of rectangles, to show the error we make and a bound of that error. We can see an animation about that.

Before considering the theoretical aspect of the problem we can think a little about the approximation of the value of the definite integral of a monotonic function.

When approximating an integral using rectangles we make an error. In some cases, for example, in monotonic functions, we can limit the magnitude of the error. The error is equal to the sum of the areas of the blue "curved triangles".

Definite integral of monotonic functions: When approximating an integral using rectangles we make an error | matematicasVisuales

This error is less than the area of a rectangle.

Definite integral of monotonic functions: The error is less than the area of a rectangle, error bound | matematicasVisuales
Definite integral of monotonic functions: The error is less than the area of a rectangle, error bound | matematicasVisuales

If we consider a simple case, when the bases of the rectangles are equal, then we can calculate an error bound:

Definite integral of monotonic functions:  when the bases of the rectangles are equal, we can calculate an error bound | matematicasVisuales

Using refined partitions we obtain as small bounds of the error as we want. Because when we refine a partition the base of the rectangle is smaller.

Definite integral of monotonic functions: refined partitions, small error bound | matematicasVisuales

Now we are going to prove that a monotonic positive increasing function on [a,b] is integrable.

We start with a partition of the interval:

Definite integral of monotonic functions: partition of an interval | matematicasVisuales
Definite integral of monotonic functions: partition of an interval | matematicasVisuales

If we divide the interval into n equal parts then the width of each rectangle is:

We can calculate an upper bound of the integral:

Definite integral of monotonic functions: upper error bound | matematicasVisuales

We consider the decreasing sequence

Definite integral of monotonic functions: a decreasing sequence | matematicasVisuales

With the same partition, this is a lower bound of the integral:

Definite integral of monotonic functions: a lower error bound | matematicasVisuales

We consider the increasing sequence

Definite integral of monotonic functions: an increasing squence | matematicasVisuales

If this two sequences converge toward one and the same limit, we can call this limit the definite integral, and write

Definite integral of monotonic functions: If this two sequences converge toward one and the same limit, we can call this limit the definite integral | matematicasVisuales

But the question always is whether they do converge.

When we consider an increasing monotonic function we have an increasing sequence (Sn) and a decreasing sequence (Tn). Then the question is only whether we can verify

Definite integral of monotonic functions | matematicasVisuales

We can calculate Tn - Sn

Definite integral of monotonic functions | matematicasVisuales

Then the limit

We conclude that if f is monotonic in the interval [a,b], then the definite integral exists. [Toeplitz, p.64]

Definite integral of monotonic functions: if f is monotonic in the interval [a,b], then the definite integral exists | matematicasVisuales

This is Newton's approach to the question in 'The Mathematical Principles of Natural Philosophy' ('Principia'). You can read this book in Google books (p. 17), California Digital Library or a latin version in Cambridge Digital Library, for example.

Definite integral of monotonic functions: Monotonic functions in Newton's Principia | matematicasVisuales

REFERENCES

Markushevich, Áreas y logaritmos. Ed. Mir.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc. (p. 77).
Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc. (p. 256)
Otto Toeplitz, The Calculus, a genetic approach, The University of Chicago Press, 1963 (p. 63).
Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer-Verlag New York Inc., 1980 (p. 190).

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