We saw previously that the area under a curve is a limit of a
sum. In general, such a limit is called a definite integral. Here is the
formal definition.
If f is a
function defined on a≤x≤b, we divide the
interval [a,b] into n subintervals [xi−1,xi] of
equal width Δx=b−an. Then the definite
integral of f from a to b is ∫baf(x)dx=limn→∞n∑i=1f(xi)Δx
provided that limit exists. If it does
exist, we say that f is integrable on [a,b].
If f(x)≥0 for all x in [a,b], then ∫baf(x)dx
represents the area under the curve y=f(x) between x=a and
x=b. But integrals make sense even when f(x) isn't positive,
as you will see in the video below.
Warning:
The definite integral of a
function f, sometimes just called the integral
of f, is not equal to an
antiderivative of f. An antiderivative of
f is a function F such that
F′=f. The definite integral of f is a number, which can be viewed as
representing the (positive and negative) area between f and
the x-axis.
However, these two different concepts, the
definite integral and the antiderivative, are related by the
beautiful Fundamental Theorem of
Calculus, which we will see in the next module.