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Unformatted text preview: 204 CHAPTER 2. CURVES when the curve is parametrized over a closed interval, this is a ﬁnite sum,
but it can be an inﬁnite (positive) series when the domain is an open
interval. Notice that a reparametrization of C is related to the original one
via a strictly monotone, continuous function, and this associates to every
partition of the original domain a partition of the reparametrized domain
involving the same segments of the curve, and hence having the same value
→
of ℓ (P , − ). Furthermore, when the parametrization is regular, the sum
p
above can be rewritten as a single (possibly improper) integral. This shows
Remark 2.5.3. The arclength of a parametrized curve C does not change
under reparametrization. If the curve is regular, then the arclength is given
by the integral of the speed (possibly improper if the domain is open)
b s (C ) = ds
a
b =
a
b =
a ds
dt dt →
− (t) dt.
˙
p for any regular parametrization of C .
In retrospect, this justiﬁes our notation for speed, and also ﬁts our
intuitive notion that the length of a curve C is the distance travelled by a
point as it traverses C once.
As a ﬁnal example, we calculate the arclength of one “arch” of the cycloid
x = θ − sin θ y = 1 − cos θ or
− (θ ) = (θ − sin θ, 1 − cos θ ),
→
p 0 ≤ θ ≤ 2π. Diﬀerentiating, we get
− (θ ) = (1 − cos θ, sin θ )
→
v
so
ds =
= (1 − cos θ )2 + sin2 θ dθ √
2 − 2 cos θ dθ. ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

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