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

546 Chapter 8 Further Applications of Integration
As a check on our answer to Example 1,
notice from Figure 5 that the arc length
ought to be slightly larger than the
distance from s1, 1d to s4, 8d, which is
s58 < 7.615773
According to our calculation in
Example 1, we have
L − 1 27 (80 s10 2 13s13 )
< 7.633705
Sure enough, this is a bit greater than
the length of the line segment.
Therefore
L − 4 9 y 10
10
13y4 s u du − 4 9 ? 2 3 u 3y2 g 13y4
− 8
27 f10 3y2 2 ( 13
4 ) 3y2 g − 1
27 (80s10 2 13s13 ) n
If a curve has the equation x − tsyd, c < y < d, and t9syd is continuous, then by
interchanging the roles of x and y in Formula 2 or Equation 3, we obtain the following
formula for its length:
4
L − y d
s1 1 d
ft9sydg2 dy −
c
y Î1 1S dx dy
dyD2
c
Example 2 Find the length of the arc of the parabola y 2 − x from s0, 0d to s1, 1d.
SOLUtion Since x − y 2 , we have dxydy − 2y, and Formula 4 gives
1
L − y Î1 1S dx 2
dy −
0 dyD y 1
s1 1 0 4y2 dy
We make the trigonometric substitution y − 1 2 tan , which gives dy − 1 2 sec2 d and
s1 1 4y 2 − s1 1 tan 2 − sec . When y − 0, tan − 0, so − 0; when y − 1,
tan − 2, so − tan 21 2 − , say. Thus
L − y
sec ? 1 2
0 sec2 d − 1 2 y
0 sec3 d
− 1 2 ? 1 2 fsec tan 1 ln | sec 1 tan
|g 0
− 1 4 ssec tan 1 ln | sec 1 tan |d
(from Example 7.2.8)
Figure 6 shows the arc of the parabola
whose length is computed in Example 2,
together with polygonal approximations
having n − 1 and n − 2 line segments,
respectively. For n − 1 the approximate
length is L 1 − s2 , the diagonal of a
square. The table shows the approximations
L n that we get by dividing f0, 1g
into n equal subintervals. Notice that
each time we double the number of sides
of the polygon, we get closer to the
exact length, which is
L − s5 ln( 2 1 s5 1 2)
< 1.478943
4
FIGURE 6
(We could have used Formula 21 in the Table of Integrals.) Since tan − 2, we have
sec 2 − 1 1 tan 2 − 5, so sec − s5 and
y
1
x=¥
L − s5
2 1 lnss5 1 2d
4
0
1
x
n
L n
1 1.414
2 1.445
4 1.464
8 1.472
16 1.476
32 1.478
64 1.479
n
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