James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 8.1 Arc Length 547
Because of the presence of the square root sign in Formulas 2 and 4, the calculation of
an arc length often leads to an integral that is very difficult or even impossible to evaluate
explicitly. Thus we sometimes have to be content with finding an approximation to the
length of a curve, as in the following example.
Example 3
(a) Set up an integral for the length of the arc of the hyperbola xy − 1 from the
point s1, 1d to the point s2, 1 2 d.
(b) Use Simpson’s Rule with n − 10 to estimate the arc length.
SOLUTION
(a) We have
dy
dx − 2 1 x 2
y − 1 x
and so the arc length is
2
L − y Î1 1S dy 2 2
dx −
1 dxD y Î1 1 1
1 x dx − y 2
4 1
sx 4 1 1
x 2 dx
(b) Using Simpson’s Rule (see Section 7.7) with a − 1, b − 2, n − 10, Dx − 0.1, and
f sxd − s1 1 1yx 4 , we have
Checking the value of the definite integral
with a more accurate approximation
produced by a computing device,
we see that the approximation using
Simpson’s Rule is accurate to four
decimal places.
L − y
2
< Dx
3
1
Î1 1 1 x 4 dx
f f s1d 1 4 f s1.1d 1 2 f s1.2d 1 4 f s1.3d 1 ∙ ∙ ∙ 1 2 f s1.8d 1 4 f s1.9d 1 f s2dg
< 1.1321 n
The Arc Length Function
We will find it useful to have a function that measures the arc length of a curve from a particular
starting point to any other point on the curve. Thus if a smooth curve C has the
equation y − f sxd, a < x < b, let ssxd be the distance along C from the initial point
P 0 sa, f sadd to the point Qsx, f sxdd. Then s is a function, called the arc length function,
and, by Formula 2,
5
ssxd − y x
s1 1 f f 9stdg 2 dt
a
(We have replaced the variable of integration by t so that x does not have two meanings.)
We can use Part 1 of the Fundamental Theorem of Calculus to differentiate Equation 5
(since the integrand is continuous):
ds
dx − s1 1 f f 9sxdg2 −Î1 1S dxD
dy 2
6
Equation 6 shows that the rate of change of s with respect to x is always at least 1 and is
equal to 1 when f 9sxd, the slope of the curve, is 0. The differential of arc length is
7
ds −Î 1 1 S dy
dxD2
dx
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.