Table 5.4.1 lists arc-length formulas for curves given explicitly, and parametrically. The arc-length function , defined by an appropriate integral, gives the length of a curve from some initial point to a varying endpoint. The differential of this function is , the arc-length element.
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Curve
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Arc-Length Element ()
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Arc-Length Function
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Table 5.4.1 Arc-length formulas
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When a limit of integration in a definite integral is a variable, that variable must not coincide with the independent variable in the integrand. That is why , , and are used in the definitions of the arc-length functions.
Because of the square root appearing in arc-length integrals, these integrals are often extremely difficult, it not impossible, to evaluate in terms of elementary functions.
A word on grammar: When used as a noun, "arc length" is typically not hyphenated. However, when used as an adjective, as in "arc-length function", this text employs the hyphen.
