1.3.4 Arc length
- Author:
- Sarah Harrelson
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:
This you can now hopefully recognize as an integral (an infinite sum of rectangles - here height times base is ) allowing us to conclude:
We can now extend this observation to something we'll call arc length of a path. If is differentiable (but not necessarily injective) then we define the arc length of as:
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.