We can put together the last few activities to conclude that the length of a curve an be computed via a differentiable injective parameterization as follows:[br][br][math]L\left(C\right)=\lim_{n\to\infty}\sum\left|\left|\vec{c}'\left(t_i^{\cdot}\right)\right|\right|\Delta t[/math][br][br]This you can now hopefully recognize as an integral (an infinite sum of rectangles - here height times base is [math]\left|\left|\vec{c}'\left(t_i^{\cdot}\right)\right|\right|\Delta t[/math]) allowing us to conclude:[br][br][math]L\left(C\right)=\int_a^b\left|\left|\vec{c}'\left(t\right)\right|\right|dt[/math][br][br]We can now extend this observation to something we'll call [b][color=#ff0000]arc length of a path[/color][/b]. If [math]\vec{c}:\left[a,b\right]\to\mathbb{R}[/math] is differentiable (but not necessarily injective) then we define the arc length of [math]\vec{c}[/math] as:[br][math]s\left(\vec{c}\right)=\int_a^b\left|\left|\vec{c}'\left(t\right)\right|\right|dt[/math][br][br]With this new definition, arclength can be thought of as a measure of the total distance the particle tracing out a curve has traveled. In the GeoGebra applet below you can type in a path and its domain of definition. The animation will show you the curve being traced out by the path together with the arclength drawn as a graph over time.