Application: Arc Length
7.1The General Problem
The Riemann integral has a wide variety of applications. In this section, using the
‘subdivide and conquer’ strategy we will show how it can be used to determine
the lengths of certain curves.
EXAMPLE 7.1.Verizon is hanging fiber optic cable around Geneva. It wants to know
how much cable it will need. Since the cable must be hung with some slack (why?),
the engineers can’t simply measure the distance between the poles to know how much
cable to use.
Figure 7.1: A hanging chain forms a
catenary. http://en.wikipedia.org/
wiki/Catenary
The curve that an idealized hanging chain or cable assumes when supported at its
ends and acted on only by its own weight is called a catenary. The equation for the
graph of a catenary curve is a hyperbolic cosine function
acosh x
a=a
2(ex/a+e−x/a),
where ais a constant. How can Verizon use this equation to find the length of the
cable it needs?
Let’s try to attack this problem in a more general fashion. Suppose y=f(x)is
a continuous function on the closed interval [a,b]. Find the length of the graph of
y=f(x)on this interval. This length is called the arc length of the curve.
Think about how we approached the area problem. We used the only figures for
which we had an area formula (rectangles) and used those figures to approximate
the area under a curve. Well, the only curve whose length we are certain of is a
straight line segment.1If the segment has endpoints (x1,y1)and (x2,y2), then the 1You might protest that you know the
circumference of a circle: 2πr. But why
is that formula true?
length of the segment is given by the distance formula
Segment Length =q(x2−x1)2+ (y2−y1)2.
So somehow we must use line segments (the only curves we know how to mea-
sure) to obtain the length of a more general continuous curve.
Well, let’s do the usual thing: ‘subdivide and conquer.’ Suppose fis continuous
on [a,b]. Given a ruler, we might mark off successive points Q1,Q2, . . . , Qnalong
the curve and take the resulting polygonal arc as an approximation to the curve
(see Figure 7.2). We could measure the length of each segment of the polygonal arc
with our ruler (or the distance formula) and then summing these values together
we would have an approximation of the length of the curve. This idea (and the
lack of a precise definition of length for curves) motivates the following process.
Assume that fis continuous over the closed, bounded interval [a,b]. Let P=
{x0,x1, . . . , xn}be a regular partition of [a,b]into nequal width subintervals. For
i=1 to nlet Qi= (xi,f(xi)) be the corresponding set of points on the graph
of f. Then the polygonal arc from Q0to Qnis just the sequence of line segments
