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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 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 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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