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

862 Chapter 13 Vector Functions
that its length is
2 L − y b
sf f 9stdg2 1 ft9stdg 2 1 fh9stdg 2 dt
a
b
− y dtD
dx
2
1S
dtD
dy
2
1S
dtD
dz
2
a
ÎS
Notice that both of the arc length formulas (1) and (2) can be put into the more compact
form
dt
In the next section we will see that if
rstd is the position vector of a moving
object at time t, then r9std is the velocity
vector and | r9std | is the speed. Thus
Equation 3 says that to compute distance
traveled, we integrate speed.
3 L − y b
because, for plane curves rstd − f std i 1 tstd j,
a
| r9std | dt
| r9std | − | f 9std i 1 t9std j | − sf f 9stdg2 1 ft9stdg 2
and for space curves rstd − f std i 1 tstd j 1 hstd k,
| r9std | − | f 9std i 1 t9std j 1 h9std k | − sf f 9stdg2 1 ft9stdg 2 1 fh9stdg 2
Figure 2 shows the arc of the helix
whose length is computed in
Example 1.
(1, 0, 0)
x
(1, 0, 2π)
FIGURE 2
z
y
ExamplE 1 Find the length of the arc of the circular helix with vector equation
rstd − cos t i 1 sin t j 1 t k from the point s1, 0, 0d to the point s1, 0, 2d.
Solution Since r9std − 2sin t i 1 cos t j 1 k, we have
| r9std | − ss2sin td2 1 cos 2 t 1 1 − s2
The arc from s1, 0, 0d to s1, 0, 2d is described by the parameter interval 0 < t < 2
and so, from Formula 3, we have
L − y 2
| r9std | dt − y 2
0 0
s2 dt − 2s2
A single curve C can be represented by more than one vector function. For instance,
the twisted cubic
4 r 1 std − kt, t 2 , t 3 l 1 < t < 2
■
could also be represented by the function
5 r 2 sud − ke u , e 2u , e 3u l 0 < u < ln 2
where the connection between the parameters t and u is given by t − e u . We say that
Equations 4 and 5 are parametrizations of the curve C. If we were to use Equation 3 to
com pute the length of C using Equations 4 and 5, we would get the same answer. In
general, it can be shown that when Equation 3 is used to compute arc length, the answer
is independent of the parametrization that is used.
The Arc Length Function
Now we suppose that C is a curve given by a vector function
rstd − f stdi 1 tstdj 1 hstdk
a < t < b
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