Areas, Integrals, and Antiderivatives
Areas, Integrals, and Antiderivatives
This section explores properties of functions defined as areas and examines some of the connections among areas, integrals and antiderivatives. In order to focus on the geometric meaning and connections, all of the functions in this section are nonnegative, but the results are generalized in the next section and proved true for all continuous functions. This section also introduces examples to illustrate how areas, integrals and antiderivatives can be used.
When is a continuous, nonnegative function, then the "area function"
represents the area between the graph of
, the t–axis, and between the vertical lines at
and
(Fig. 1), and the derivative of
represents the rate of change (growth) of
. Examples 2 and 3 of Section 4.3 showed that for
some functions f, the derivative of
was equal to
so
was an antiderivative of
. The next theorem says the result is true for every continuous, nonnegative function
.
The Area Function is an Antiderivative
If is a continuous nonnegative function,
and
.
then , so
is an antiderivative of
.
This result relating integrals and antiderivatives is a special case (for nonnegative functions ) of the
Fundamental Theorem of Calculus (Part 1) which is proved in Section 4.5 . This result is important for
two reasons:
(i) it says that a large collection of functions have antiderivatives, and
(ii) it leads to an easy way of exactly evaluating definite integrals.
Example 1: for the function
shown in Fig. 2.
Estimate the values of and
for
and
and use these values to sketch the graph of
.
Solution: Dividing the region into squares and triangles, it is easy to see that and
. Since
, we
know that
and
. The graph of
is shown in Fig. 3.
It is important to recognize that f is not differentiable at and
.
However, the values of
change smoothly near 2 and 3, and the function
is differentiable at those points and at every other point from 1 to 5. Also,
(
is clearly decreasing near
), but
is positive
(the area
is growing even though f is getting smaller).
Practice 1: is the area bounded by the horizontal axis, vertical lines at
and
, and the graph of
shown in Fig. 4. Estimate the values of
and
for
and
.
Example 2: Let . Evaluate
for
.
Solution: Fig. 5(a) shows the graph of , and
is the derivative of
. By the theorem,
so
, and
. Fig. 5(b) shows the graph of
and 5(c) is the graph of
.
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.5-Areas-Integrals-Antiderivatives.pdf This work is licensed under a Creative Commons Attribution 3.0 License.