Suppose a space curve [math]C[/math] is defined by component functions [math]x=g\left(t\right)[/math], [math]y=h\left(t\right)[/math], [math]z=k\left(t\right)[/math] for values of [math]t[/math] from [math]t=a[/math] to [math]t=b[/math]. Moreover, suppose that the points of [math]C[/math] lie in the domain of a function [math]f\left(x,y,z\right)[/math]. We wish to compute the [i]line integral of [/i][math]f[/math][i] over [/i][math]C[/math]. We write this as[br] [math]\int_Cf\left(x,y,z\right)ds[/math][br]where the differential [math]ds[/math] represents change along path [math]C[/math]. To compute, we can express this integral in terms of the parameter [math]t[/math]:[br] [math]\int_Cf\left(x,y,z\right)ds=\int_a^bf\left(g\left(t\right),h\left(t\right),k\left(t\right)\right)\left|\textbf{v}\left(t\right)\right|dt[/math].[br]In the above expression, [math]ds[/math] is replaced by [math]\left|\textbf{v}\left(t\right)\right|dt[/math], where [math]\left|\textbf{v}\left(t\right)\right|=\sqrt{\left(g'\left(t\right)\right)^2+\left(h'\left(t\right)\right)^2+\left(k'\left(t\right)\right)^2}[/math].

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]