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

Section 10.2 Calculus with Parametric Curves 653
The sum in (4) resembles a Riemann sum for the function sf f 9stdg 2 1 ft9stdg 2 but it is
not exactly a Riemann sum because t i * ± t i ** in general. Nevertheless, if f 9 and t9 are
contin uous, it can be shown that the limit in (4) is the same as if t i * and t i ** were equal,
namely,
L − y sf f 9stdg2 1 ft9stdg 2 dt
Thus, using Leibniz notation, we have the following result, which has the same form as
Formula 3.
5 Theorem If a curve C is described by the parametric equations x − f std,
y − tstd, < t < , where f 9 and t9 are continuous on f, g and C is traversed
exactly once as t increases from to , then the length of C is
L − yÎS dtD
dx
2
1S
dtD
dy
2
dt
Notice that the formula in Theorem 5 is consistent with the general formulas L − y ds
and sdsd 2 − sdxd 2 1 sdyd 2 of Section 8.1.
Example 4 If we use the representation of the unit circle given in Example 10.1.2,
0
ÎS
x − cos t y − sin t 0 < t < 2
then dxydt − 2sin t and dyydt − cos t, so Theorem 5 gives
2
L − y dtD
dx
2
1S
dtD
dy
2
dt − y 2
ssin 2 t 1 cos 2 t dt − y 2
dt − 2
0
0
as expected. If, on the other hand, we use the representation given in Example 10.1.3,
y 2
0
ÎS
x − sin 2t y − cos 2t 0 < t < 2
then dxydt − 2 cos 2t, dyydt − 22 sin 2t, and the integral in Theorem 5 gives
dtD
dx
2
1S
dtD
dy
2
dt − y 2
s4 cos 2 2t 1 4 sin 2 2t dt − y 2
2 dt − 4
0
0
Notice that the integral gives twice the arc length of the circle because as t increases
from 0 to 2, the point ssin 2t, cos 2td traverses the circle twice. In general, when finding
the length of a curve C from a parametric representation, we have to be careful to
ensure that C is traversed only once as t increases from to .
n
Example 5 Find the length of one arch of the cycloid x − rs 2 sin d,
y − rs1 2 cos d.
SOLUTION From Example 3 we see that one arch is described by the parameter interval
0 < < 2. Since
dx
d − rs1 2 cos d and dy
d − r sin
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