The goal of today's lesson is to understand what it means to compute the length of a curve. The GeoGebra applet below demonstrates our approach to approximating length.

We've parameterized a curve via a path [math]\vec{c}:\left[a,b\right]\to\mathbb{R}^2[/math]. Then we form a [color=#ff0000][b]regular partition[/b] [/color]of the domain. A regular partition of an interval of real numbers is a subdivision of the interval into [math]n[/math] smaller, equally sized sub-intervals. The length of each sub-interval (also referred to as the [b][color=#ff0000]mesh[/color][/b] of the partition) is notated as [math]\Delta t[/math].[br][br]The endpoints of the sub-intervals are then mapped onto the curve via the path [math]\vec{c}\left(t\right)[/math]. The straight-line distance between successive points is computed by defining something I'll call a secant vector (this is the same usage of the word secant as in Geometry (a line that cuts through a circle in two points) and AP Calculus (a line connecting two points on a graph)). We'll define the [math]i^{\text{th}}[/math] secant vector this way:[br][math]\vec{s}_i=\vec{c}\left(a+i\Delta t\right)-\vec{c}\left(a+\left(i-1\right)\Delta t\right)[/math]. Our approximation for the length of the curve is then the sum of the lengths of these secant vectors.[br][br]There are several subtleties to consider.