For curves given in different formats, Table 2.2.1 lists formulas for the arc-length function, which measures the length of the curve as a function of the curve's parameter. In each case, the integrand for the arc-length function is recognized as . Hence, by the fundamental theorem of calculus, , and again by elementary calculus, .
Curve
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Arc-Length Function
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Table 2.2.1 The arc-length function
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If the parameter for a curve is s, the curve's arc length, then (by the chain rule)
⇒ =