Let be a continuous function on interval .
Divide into subintervals of length
.
Choose (sample) points in th interval, for each .
The (signed) area between the graph of and the axis is approximately

(The is notation to make it easier to write down and think
about the sum.)

**Definition 2.1.1** (Signed Area)
The

*(signed) area between the graph* of

and the

axis between

and

is

(Note that

depends on

.)

It is a theorem that the area exists and doesn't depend
on the choice of .

William Stein
2006-03-15