Engineering Calculus Notes 225

Engineering Calculus Notes 225 - 213 2.5. INTEGRATION ALONG...

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Unformatted text preview: 213 2.5. INTEGRATION ALONG CURVES (d) Show that the sum ∞ k =1 △sk diverges. (e) Thus, if we take (for example) the piecewise linear approximations to the curve obtained by taking the straight line segments as above to some finite value of k and then join the last point to (0, 0), their lengths will also diverge as the finite value increases. Thus, there exist partitions of the curve whose total lengths are arbitrarily large, and the curve is not rectifiable. Challenge problem: 5. Bolzano’s curve (continued): We continue here our study of the curve described in Exercise 15 in § 2.4; we keep the notation of that exercise. (a) Show that the slope of each straight piece of the graph of fk has the form m = ±2n for some integer 0 ≤ n ≤ k. Note that each interval over which fk is affine has length 3−k . (b) Show that if two line segments start at a common endpoint and end on a vertical line, and their slopes are 2n and 2n+1 respectively, then the ratio of the second to the first length is m2 = m1 1 + 2n+1 1 + 2n Show that this quantity is non-decreasing, and that therefore it is always at least equal to 5/3. (c) Use this to show that the ratio of the lengths of the graphs of √√ fk+1 and fk are bounded below by 2 5/3 3 + 1/3 ≥ 1.19. (d) How does this show that the graph of f is non-rectifieable? ...
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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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