Next we want to figure out the length of a parametrized curve. As with all integrals, we break it into pieces, estimate each piece, add the pieces together, and take a limit.

In the following video we derive the formulas for these quantities, compute the arclength of a cycloid, and compute the surface area of the associated surface of revolution.



Summary: If we have a parametrized curve running from time $t_1$ to time $t_2$, then
  1. The arc length of the curve is $$L = \int_{t_1}^{t_2} \sqrt{\left ( \frac{dx}{dt}\right )^2 + \left ( \frac{dy}{dt}\right )^2} \;dt,$$
  2. If we rotate the curve around the $x$ axis we get a surface, called a surface of revolution. The area of this surface is $$\int_{t_1}^{t_2} 2 \pi y \; \sqrt{\left ( \frac{dx}{dt}\right )^2 + \left ( \frac{dy}{dt}\right )^2} \;dt.$$