93
Section 3.8
.
Applications of Calculus
3.8
.
Applications of Calculus
In the Applications of Calculus section at the end of the previous
chapter, I discussed how the slope-of-the-tangent-line idea related to the
calculus concept of a derivative, and the branch of calculus known as
differential calculus. The other main branch of calculus, integral calculus,
has to do with the area-under-the-curve concept discussed in section 3.5 of
this chapter. Again there is a concept, a notation, and a bag of tricks for
doing things symbolically rather than graphically. In calculus, the area
under the v-t graph between t=t
1
and t=t
2
is notated like this:
areaunderthecurve=
.
x=vdt
t1
t2
The expression on the right is called an integral, and the s-shaped symbol,
the integral sign, is read as “integral of....”
Integral calculus and differential calculus are closely related. For in-
stance, if you take the derivative of the function x(t), you get the function
v(t), and if you integrate the function v(t), you get x(t) back again. In other
words, integration and differentiation are inverse operations. This is known
as the fundamental theorem of calculus.
On an unrelated topic, there is a special notation for taking the deriva-
tive of a function twice. The acceleration, for instance, is the second (i.e.
double) derivative of the position, because differentiating x once gives v, and
then differentiating v gives a. This is written as
a=d
2
x
dt
2
.
The seemingly inconsistent placement of the twos on the top and bottom
confuses all beginning calculus students. The motivation for this funny
notation is that acceleration has units of m/s
2
, and the notation correctly
suggests that: the top looks like it has units of meters, the bottom seconds
2
.
The notation is not meant, however, to suggest that t is really squared.