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Engineering Calculus Notes 225

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

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2.5. INTEGRATION ALONG CURVES 213 (d) Show that the sum summationdisplay k =1 s k 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 f k has the form m = ±
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