Applied Calculus Math 215 - University of Hawaii

6 CHAPTER 0. A PREVIEW
0.25
0.2
0.15
0.1
0.05
R(t)
1 2 3 4 5 6 t
Figure 6: Time dependent rate
0.7
0.6
0.5
0.4
0.3
0.2
0.1
A(t)
1 2 3 4 5 6 t
Figure 7: Amount absorbed
medication which has been absorbed between t =0andt=T is the area
under the graph of R(t) between t =0andt=T. We denote this function
by A(T ). Using methods which you will learn in this course, we found the
function A. The graph is shown in Figure 7. You may find the value for
A(4) in the graph. A numerical calculation yields A(4) = 0.6735.
More generally, one may want to find the area under the graph of a
function f(x) between x = a and x = b. To make sense out of this we first
need to clarify what we mean when we talk about the area of a region, in
particular if the region is not bounded by straight lines. Next we need to
determine the areas of such regions. In fact, finding the area between the
graph of a non-negative function f and the x-axis between x = a and x = b
means to integrate f from a to b. Both topics are addressed in the chapter
on integration.
The ideas of differentiation and integration are related to each other. If
we differentiate the function shown in Figure 7 at some time t, thenweget
the function in Figure 6 at t. You will understand this after the discussion
in Section 4.6. In this section we also discuss the Fundamental Theorem of
Calculus, which is our principal tool to calculate integrals.
The two basic ideas of the rate of change of a function and the area
below the graph of a function will be developed into a substantial body
of mathematical results that can be applied in many situations. You are
expected to learn about them, so you can understand other sciences where
they are applied.