Unit 8
Applications of Integration
10–15% AB | 6–9% BCCED Unit 8
Accumulation & Applications
Average value, motion (displacement vs. distance), and net change in context.
Average Value of a Function
Average Value
The average value is the height of a rectangle with the same area as the region under the curve.
⚠ Watch Out
Don't forget the 1/(b − a) factor
Average value = (1/(b−a)) ∫f(x) dx, not just ∫f(x) dx. Divide by the interval length.Displacement (Net Change)
Integral of Velocity
Can be positive, negative, or zero. Where you end up relative to start.
Total Distance Traveled
Integral of Speed
Always positive. How far you actually moved.
Net Change Theorem
The Integral of a Rate Gives Net Change
| If F'(x) represents... | Then ∫F'(x) dx gives... |
|---|---|
| Rate of water flowing (gal/min) | Total gallons added/removed |
| Rate of population change (people/year) | Net change in population |
| Velocity (ft/sec) | Displacement (ft) |
◉ Observe
Connection to Unit 4: Same motion concepts, but now using integrals instead of derivatives. Position = ∫velocity. Velocity = ∫acceleration.
AP® Calculus AB & BC · Unit 8 Overview · Mr. Brantley