AP Calculus BC / Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.3 Finding Arc Lengths of Curves Given by Parametric Equations
Notes
$$L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$$
- For a particle, this is the total distance traveled.
Practice
Arc length of $x=3t$, $y=4t$ for $0\le t\le2$:
$\displaystyle \int_0^2 5\,dt$.
$x=\cos t$, $y=\sin t$ on $[0,2\pi]$ has length:
The integrand is 1: a full unit circle.
The parametric arc length integrand is the same as:
$\displaystyle \sqrt{x'^2+y'^2}$ is speed.
Ready for more? Take the Unit 9 test.