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

652 Chapter 10 Parametric Equations and Polar Coordinates
Arc Length
We already know how to find the length L of a curve C given in the form y − Fsxd,
a < x < b. Formula 8.1.3 says that if F9 is continuous, then
2
b
L − y aÎ S 1 1 dy dx
dxD2
Suppose that C can also be described by the parametric equations x − f std and y − tstd,
< t < , where dxydt − f 9std . 0. This means that C is traversed once, from left to
right, as t increases from to and f sd − a, f sd − b. Putting Formula 1 into Formula
2 and using the Substitution Rule, we obtain
L − y
b
a
Î
S
dxD
1 1 dy
2
dx − y Î
S dxydtD
1 1 dyydt 2
dx
dt dt
Since dxydt . 0, we have
3
L − y ÎS
dtD
dx
2
1S
dtD
dy
2
dt
y
P
P¡
0
FIGURE 4
C
P¸
P i_1
P i
P n
x
Even if C can’t be expressed in the form y − Fsxd, Formula 3 is still valid but we
obtain it by polygonal approximations. We divide the parameter interval f, g into n
subintervals of equal width Dt. If t 0 , t 1 , t 2 , . . . , t n are the endpoints of these subintervals,
then x i − f st i d and y i − tst i d are the coordinates of points P i sx i , y i d that lie on C and the
polygon with vertices P 0 , P 1 , . . . , P n approximates C. (See Figure 4.)
As in Section 8.1, we define the length L of C to be the limit of the lengths of these
approximating polygons as n l `:
L − lim
nl ` on
i−1 | P i21 P i |
The Mean Value Theorem, when applied to f on the interval ft i21 , t i g, gives a number t i *
in st i21 , t i d such that
f st i d 2 f st i21 d − f 9st i *dst i 2 t i21 d
If we let Dx i − x i 2 x i21 and Dy i − y i 2 y i21 , this equation becomes
Dx i − f 9st i *d Dt
Similarly, when applied to t, the Mean Value Theorem gives a number t i ** in st i21 , t i d
such that
Dy i − t9st i **d Dt
Therefore
and so
4
| P i21P i | − ssDx id 2 1 sDy i d 2 − sf f 9st i *dDtg 2 1 ft9st i **dDtg 2
− sf f 9st i *dg 2 1 ft9st i **dg 2 Dt
L − lim
n l ` o n
sf f 9st i *dg 2 1 ft9st i **dg 2 Dt
i−1
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