Engineering Calculus Notes 216

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

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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.

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