3.3 Arc Length and Curvature - Calculus Volume 3 | OpenStax

Arc length of the curve is given by $s = \int _{t_1} ^{t_2} \sqrt{(x’(t)^2+(y’(t)^2}dt$

• Arc length function is given by $s = \int _a ^b ||\vec r’(u)||du$
  • Can be used to find the arc-length parameterization of a curve
  • Bounds of integration are in terms of $t$
• In polar coordinates, $L = \int _{\alpha} ^{\beta} \sqrt{r^2+ r’(\theta)^2}d \theta$
• Principal unit tangent vector is $\vec T(t) = \frac{\vec r(t)}{||\vec r(t)||}$