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:

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.

monotonic function from an interval to . Let be a non-positive number such that for all . Let .

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

In general you should think of as the difference where

and

Which is larger:

- a)
- or ?
- b)
- or ?
- c)
- or ?
- d)
- or ?
- e)
- or ?

Graph of

- a)
- .
- b)
- .
- c)
- .

- a)
- .
- b)
- .
- c)
- .
- d)
- .