1.2 Calculus of parametric curves  (Page 3/6)

Then a Riemann sum for the area is

Multiplying and dividing each area by $t_i-t_{i-1}$ gives

Taking the limit as $n$ approaches infinity gives

This leads to the following theorem.

Arc length of a parametric curve

In addition to finding the area under a parametric curve, we sometimes need to find the arc length of a parametric curve. In the case of a line segment, arc length is the same as the distance between the endpoints. If a particle travels from point A to point B along a curve, then the distance that particle travels is the arc length. To develop a formula for arc length, we start with an approximation by line segments as shown in the following graph.

Given a plane curve defined by the functions $x=x\left(t\right),y=y\left(t\right),a\le t\le b,$ we start by partitioning the interval $[a,b]$ into n equal subintervals: $t_0=a<t_1<t_2<\text{⋯}<t_n=b.$ The width of each subinterval is given by $\text{Δ}t=(b-a)\text{/}n.$ We can calculate the length of each line segment:

Then add these up. We let s denote the exact arc length and $s_n$ denote the approximation by n line segments:

If we assume that $x\left(t\right)$ and $y\left(t\right)$ are differentiable functions of t, then the Mean Value Theorem ( Introduction to the Applications of Derivatives ) applies, so in each subinterval $[t_{k-1},t_k]$ there exist ${\hat{t}}_k$ and ${\tilde{t}}_k$ such that

Therefore [link] becomes

This is a Riemann sum that approximates the arc length over a partition of the interval $[a,b].$ If we further assume that the derivatives are continuous and let the number of points in the partition increase without bound, the approximation approaches the exact arc length. This gives

When taking the limit, the values of ${\hat{t}}_k$ and ${\tilde{t}}_k$ are both contained within the same ever-shrinking interval of width $\text{Δ}t,$ so they must converge to the same value.