204CHAPTER 2. CURVESwhen 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 openinterval. Notice that a reparametrization ofCis related to the original onevia a strictly monotone, continuous function, and this associates to everypartition of the original domain a partition of the reparametrized domaininvolving the same segments of the curve, and hence having the same valueofℓ(P,−→p). Furthermore, when the parametrization is regular, the sumabove can be rewritten as a single (possibly improper) integral. This showsRemark 2.5.3.The arclength of a parametrized curveCdoes not changeunder reparametrization. If the curve is regular, then the arclength is givenby the integral of the speed (possibly improper if the domain is open)
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