Mathematics

This is about arc length, which measures the total distance traveled along a curve. The key insight is that arc length isn't just the straight-line distance between endpoints—you need to account for how fast the position changes at each instant. The formula $L = \int_a^b |\mathbf{r}'(t)| \, dt$ sums up all those infinitesimal distances by integrating the speed (magnitude of the velocity vector) over the time interval.

This is about arc length, which measures the total distance traveled along a curve. The key insight is that arc length isn't just the straight-line distance between endpoints—you need to account for how fast the position changes at each instant. The formula $L = \int_a^b |\mathbf{r}'(t)| \, dt$ sums up all those infinitesimal distances by integrating the speed (magnitude of the velocity vector) over the time interval.