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Engineering Calculus Notes 216

Engineering Calculus Notes 216 - 204 CHAPTER 2 CURVES when...

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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 ( P , −→ p ). Furthermore, when the parametrization is regular, the sum 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)
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