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

548 Chapter 8 Further Applications of Integration
y
0 x
FIGURE 7
y
1
ds
Îs
dx
P¸
1
y=≈- 8 ln x
0 1 x x
FIGURE 8
y
1
s(x)
Îy
dy
Figure 8 shows the interpretation of
the arc length function in Example 4.
Figure 9 shows the graph of this arc
length function. Why is ssxd negative
when x is less than 1?
0 1
x
FIGURE 9
1
s(x)=≈+ 8 ln x-1
and this equation is sometimes written in the symmetric form
8
sdsd 2 − sdxd 2 1 sdyd 2
The geometric interpretation of Equation 8 is shown in Figure 7. It can be used as a mnemonic
device for remembering both of the Formulas 3 and 4. If we write L − y ds, then
from Equation 8 either we can solve to get (7), which gives (3), or we can solve to get
which gives (4).
ds −Î 1 1 S dx
dyD2
dy
Example 4 Find the arc length function for the curve y − x 2 2 1 8 ln x taking P 0s1, 1d
as the starting point.
SOLUTION If f sxd − x 2 2 1 8 ln x, then
f 9sxd − 2x 2 1
8x
1 1 f f 9sxdg 2 − 1 1S2x 2
8xD
1 2
− 1 1 4x 2 2 1 2 1 1
64x 2
s1 1 f f 9sxdg 2 − 2x 1 1
8x
Thus the arc length function is given by
− 4x 2 1 1 2 1 1
64x − S2x 1 1 2
8xD
2
ssince x . 0d
ssxd − y x
s1 1 f f 9stdg 2 dt
1
− y
x
1
S2t 1 1 8tD dt − t 2 1 1 8 ln tg 1
− x 2 1 1 8 ln x 2 1
For instance, the arc length along the curve from s1, 1d to s3, f s3dd is
ss3d − 3 2 1 1 8
ln 3 2 1 − 8 1
ln 3
8 < 8.1373 n
x
1. Use the arc length formula (3) to find the length of the curve
y − 2x 2 5, 21 < x < 3. Check your answer by noting that
the curve is a line segment and calculating its length by the
distance formula.
2. Use the arc length formula to find the length of the curve
y − s2 2 x 2 , 0 < x < 1. Check your answer by noting that
the curve is part of a circle.
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