AP Calculus AB and BC glossary
Total distance
Total distance traveled is the integral of the absolute value of velocity, which is the integral of speed. Unlike displacement it counts motion in both directions positively, so it is always at least as large as the size of the displacement.
To evaluate it by hand, find where velocity changes sign, split the integral at those times, and add the absolute values of the pieces. On a calculator-active question the absolute value can go straight into the integrand.
The mistake
Integrating velocity and taking the absolute value at the end. The absolute value has to be inside the integral, or the legs cancel before you can stop them.
Appears in: Unit 4: Contextual Applications, Unit 8: Applications of Integration