Table 5.4.1 lists arc-length formulas for curves given explicitly, and parametrically. The arc-length function $s$, 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 $\mathrm{ds}$, the arc-length element.

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 $\stackrel{\^}{x}$, $\stackrel{\^}{y}$, and $\stackrel{\^}{t}$ 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.