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A. $\operatorname{Let} \alpha: I \rightarrow R^{3}$ be a curve of class $C^{0}$ (cf. Exercise 7 ). Use the approximation by polygons described in Exercise 8 to give...

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A. $\operatorname{Let} \alpha: I \rightarrow R^{3}$ be a curve of class $C^{0}$ (cf. Exercise 7 ). Use the approximation by polygons described in Exercise 8 to give a reasonable definition of arc length of $\alpha$ b. (A Nonrectifiable Curve.) The following example shows that, with any reasonable definition, the arc length of a $C^{0}$ curve in a closed interval may be unbounded. Let $\alpha:[0,1] \rightarrow R^{2}$ be given as $\alpha(t)=$ $(t, t \sin (\pi / t))$ if $t \neq 0,$ and $\alpha(0)=(

a. $\operatorname{Let} \alpha: I \rightarrow R^{3}$ be a curve of class $C^{0}$ (cf. Exercise 7 ). Use the approximation by polygons described in Exercise 8 to give a reasonable definition of arc length of $\alpha$ b. (A Nonrectifiable Curve.) The following example shows that, with any reasonable definition, the arc length of a $C^{0}$ curve in a closed interval may be unbounded. Let $\alpha:[0,1] \rightarrow R^{2}$ be given as $\alpha(t)=$ $(t, t \sin (\pi / t))$ if $t \neq 0,$ and $\alpha(0)=(0,0) .$ Show, geometrically, that the arc length of the portion of the curve corresponding to $1 /(n+1) \leq$ $t \leq 1 / n$ is at least $2 /\left(n+\frac{1}{2}\right) .$ Use this to show that the length of the curve in the interval $1 / N \leq t \leq 1$ is greater than $2 \sum_{n=1}^{N} 1 /(n+1)$ and thus it tends to infinity as $N \rightarrow \infty$



Answers

Use the arc length formula to find the length of the curve $ y = \sqrt{2 - x^2} $, $ 0 \le x \le 1 $. Check your answer by noting that the curve is part of a circle.


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