Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) v(t) dt means the definite integral of v(t) from t = a to t = b.
Recap
- Differentiating goes x → v → a. Integrating goes back, a → v → x, but gives only the change, so you need a starting value.
- Displacement = ∫ (a to b) v(t) dt = x(b) − x(a). Signed: backward motion cancels forward motion.
- Total distance = ∫ (a to b) |v(t)| dt. Never negative.
- By hand: split at the times where v changes sign, integrate v on each piece, take absolute values, add. With a calculator: integrate |v| directly.
- On a velocity graph, area above the t-axis is forward motion and area below is backward motion.
Key relationships
| Want | Use | Units (m, s) |
|---|---|---|
| Position at time b | x(b) = x(a) + ∫ (a to b) v(t) dt | m |
| Velocity at time b | v(b) = v(a) + ∫ (a to b) a(t) dt | m/s |
| Displacement | ∫ (a to b) v(t) dt | m |
| Total distance | ∫ (a to b) |v(t)| dt | m |
| Average velocity | (1/(b − a)) ∫ (a to b) v(t) dt | m/s |
| Average speed | (total distance)/(b − a) | m/s |
Assumptions behind the method
- Motion is along a straight line, with a chosen positive direction.
- v and a are continuous on the interval, so the Fundamental Theorem applies.
- Direction changes happen only where v changes sign.
Mistakes to avoid
- Calling ∫ v dt the distance. It is displacement.
- Writing |∫ v dt| for distance. The absolute value goes inside.
- Forgetting x(a) when asked for a position.
- Splitting where a = 0 instead of where v changes sign.
- Average speed from displacement instead of distance.
- Wrong units: ∫ a dt is a velocity, not a distance.
Quick self-check
- v(t) = 4 − 2t on [0, 3]. Find the displacement and the total distance. (Displacement 3; distance 5: 4 forward on [0, 2], then 1 back on [2, 3])
- x(0) = 5 and ∫ (0 to 2) v(t) dt = −7. Find x(2). (−2)
- v(0) = 3 m/s and a(t) = 2 m/s² for all t. Find v(5). (13 m/s = 3 + 2 × 5)
Next: practice questions.