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Engineering Calculus Notes 210

Engineering Calculus Notes 210 - 198 CHAPTER 2 CURVES and...

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Unformatted text preview: 198 CHAPTER 2. CURVES and so the supremum of ℓ ( P ′ , −→ q ) over all partitions P ′ of the domain of −→ q is at least the same as that over partitions of the domain of −→ p . Reversing the roles of the two parametrizations, we see that the two suprema are actually the same. This formulation of arclength has the advantage of being clearly based on the geometry of the curve, rather than the parametrization we use to construct it. However, as a tool for computing the arclength, it is as useful (or useless) as the definition of the definite integral via Riemann sums is for calculating definite integrals. Fortunately, for regular curves, we can use definite integrals to calculate arclength. Theorem 2.5.1. Every regular arc is rectifiable, and if −→ p : [ a,b ] → R 3 is a regular one-to-one function with image C , then s ( C ) = integraldisplay b a vextenddouble vextenddouble vextenddouble ˙ −→ p ( t ) vextenddouble vextenddouble vextenddouble dt....
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