## The definition of area under curve

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