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Unformatted text preview: 8.1: Arc Length (Dated: November 8, 2011) THE ARC LENGTH FORMULA Let C be a curve whose defining equation is y = f ( x ), where f is a continuous function on [ a , b ]. A polygonal approximation to C can be obtained by dividing the interval [ a , b ] into n subintervals of equal width Δ x = b a n . Let a = x , x 1 , x 2 , ··· , x n = b be the endpoints of the subintervals. Let P i = ( x i , y i ), where y i = f ( x i ) for each i , i = 1,2, ··· , n . Clearly, the points P i ’s lie on the curve C , from which it fol lows that the (open) polygon with vertices P , P 1 , ··· , P n is an approximation of C . Thus the length L of C satisfies the approximation L ≈ n X i = 1  P i 1 P i  , where  P i 1 P i  denotes the length of the line segment P i 1 P i , i = 1,2, ··· , n . The approximation gets better as Δ x → 0 or equivalently as n → ∞ . Hence we define the length L of the curve C as L = lim n →∞ n X i = 1  P i 1 P i  provided the limit exists....
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 Fall '09
 GELINE
 Calculus, Geometry, Approximation, Arc Length, Fundamental Theorem Of Calculus, dx, dy

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