8.2 Motion with Integrals

Displacement vs. distance

Displacement over [a, b] is ∫[a to b] v(t) dt (net change in position). Total distance is ∫[a to b] |v(t)| dt — the absolute value counts all motion as positive.

Recovering position & velocity

Final position = initial position + ∫ v dt, and integrating acceleration gives the change in velocity.

Key terms

Displacement

∫ v(t) dt (net change in position).

Total distance

∫ |v(t)| dt.

Final position

Initial position + ∫ v dt.

∫ acceleration

Gives change in velocity.

Why |v|

Counts all motion as positive for distance.

Try an AP-style question

v(t) = t² − 4 on [0, 3]. What is the particle's displacement?

1. −3
2. 23/3
3. 3
4. −16/3