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# 8.1 Definition of the Integral

If is a monotonic function from an interval to , then we have shown that for every sequence of partitions on such that , and every sequence such that for all is a sample for , we have

8.1   Definition (Integral.) Let be a bounded function from an interval to . We say that is integrable on if there is a number such that for every sequence of partitions on such that , and every sequence where is a sample for

If is integrable on then the number just described is denoted by and is called  the integral from to of .'' Notice that by our definition an integrable function is necessarily bounded.

The definition just given is essentially due to Bernhard Riemann(1826-1866), and first appeared around 1860[39, pages 239 ff]. The symbol was introduced by Leibniz sometime around 1675[15, vol 2, p242]. The symbol is a form of the letter s, standing for sum (in Latin as well as in English.) The practice of attaching the limits and to the integral sign was introduced by Joseph Fourier around 1820. Before this time the limits were usually indicated by words.

We can now restate theorems 7.6 and 7.15 as follows:

8.2   Theorem (Monotonic functions are integrable I.) If is a monotonic function on an interval with non-negative values, then is integrable on and

8.3   Theorem (Integrals of power functions.) Let , and let be real numbers such that . Let for . Then

In general integrable functions may take negative as well as positive values and in these cases does not represent an area.

The next theorem shows that monotonic functions are integrable even if they take on negative values.

8.4   Example (Monotonic functions are integrable II.) Let be a
monotonic function from an interval to . Let be a non-positive number such that for all . Let .
Then is a monotonic function from to . Hence by theorem 7.6, is integrable on and . Now let be a sequence of partitions of such that , and let be a sequence such that for each in , is a sample for . Then
 (8.5)

If and then

Thus by (8.5)

If we use the fact that , and then use the sum theorem for limits of sequences, we get

It follows from the definition of integrable functions that is integrable on and

Thus in figure b, represents the shaded area with the area of the thick box subtracted from it, which is the same as the area of the region marked " in figures c and d, with the area of the region marked " subtracted from it.

The figure represents a geometric interpretation for a Riemann sum. In the figure

is the area of and

is the negative of the area of

In general you should think of as the difference where

and

8.6   Exercise. The graphs of two functions from to are sketched below.
Let

Which is larger:
a)
or ?
b)
or ?
c)
or ?
d)
or ?
e)
or ?
Explain how you decided on your answers. Your explanations may be informal, but they should be convincing.

8.7   Exercise. Below is the graph of a function . By looking at the graph of estimate the following integrals. (No explanation is necessary.)

Graph of

a)
.
b)
.
c)
.

8.8   Exercise. Sketch the graph of one function satisfying all four of the following conditions.
a)
.
b)
.
c)
.
d)
.

Next: 8.2 Properties of the Up: 8. Integrable Functions Previous: 8. Integrable Functions   Index
Ray Mayer 2007-09-07