We looked at some applications of integrals in Chapter 6:
areas, volumes, work, and average values. Here we
explore some of the many other geometric applications of
integration—the length of a curve, the area of a surface—
as well as quantities of interest in physics, engineering,
biology, economics, and statistics. For instance, we will investigate the center of
gravity of a plate, the force exerted by water pressure on a dam, the flow of blood
from the human heart, and the average time spent on hold during a telephone call.
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9.1
Arc Length
What do we mean by the length of a curve? We might think of fitting a piece of string to
the curve in Figure 1 and then measuring the string against a ruler. But that might be
difficult to do with much accuracy if we have a complicated curve. We need a precise
definition for the length of an arc of a curve, in the same spirit as the definitions we devel-
oped for the concepts of area and volume.
If the curve is a polygon, we can easily find its length; we just add the lengths of the
line segments that form the polygon. (We can use the distance formula to find the distance
between the endpoints of each segment.) We are going to define the length of a general
curve by first approximating it by a polygon and then taking a limit as the number of seg-
ments of the polygon is increased. This process is familiar for the case of a circle, where
the circumference is the limit of lengths of inscribed polygons (see Figure 2).
Now suppose that a curve
is defined by the equation
, where
f
is continuous
and
. We obtain a polygonal approximation to
by dividing the interval
into
n
subintervals with endpoints
and equal width
. If
, then
the point
lies on
and the polygon with vertices
,
, . . . ,
, illustrated in
Figure 3, is an approximation to
. The length
L
of
is approximately the length of this
polygon and the approximation gets better as we let
n
increase. (See Figure 4, where the
arc of the curve between
and
has been magnified and approximations with succes-
sively smaller values of
are shown.) Therefore, we define the
length
of the curve
FIGURE 3
FIGURE 4
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P
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i
y
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P™
P
i-1
P
i
P
n
y=ƒ
0
x
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i
¤
x
i-1
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C
L
x
P
i
P
i
1
C
C
P
n
P
1
P
0
C
P
i
x
i
,
y
i
y
i
f x
i
x
x
0
,
x
1
, . . . ,
x
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a
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C
a
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C
583
FIGURE 1
FIGURE 2
5E-09(pp 582-591)
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6:20 PM
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