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is necessary. Because of the differentiation in the arc-length formula, the integrand's Taylor series loses one order of precision, relative to the arc itself, with a corresponding loss of precision. But for infinitely smooth functions, numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as The interval corresponds to a

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