AP Calculus AB and BC glossary

Displacement

Also called: Total distance travelled

Displacement is the net change in position over a time interval, computed by integrating velocity. Total distance travelled integrates the absolute value of velocity instead, so it counts backtracking as additional distance.

displacement=∫abv(t) dt,distance=∫ab∣v(t)∣ dt\text{displacement} = \int_a^b v(t)\,dt, \qquad \text{distance} = \int_a^b |v(t)|\,dt

To compute total distance by hand, find where velocity changes sign, split the integral at those times, and add the absolute values of the pieces. On a calculator you can integrate the absolute value directly.

The mistake

Reporting displacement when the question asks for distance travelled. If the velocity changes sign anywhere on the interval, the two answers differ.

Appears in: Unit 8: Applications of Integration