Short on time? This page is the recap. For explanations, the rate graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) r(t) dt means the definite integral of r(t) from t = a to t = b.
Recap
- A rate times a short time is a small change. Adding the small changes gives an integral, so the integral of a rate is the net change in the amount.
- Net means gains and losses cancel. For the total of all rises and falls, integrate |rate|.
- The amount at time x is the starting value plus the accumulated change. Without the starting value you only have the change.
- The derivative of the amount function is the rate (Fundamental Theorem).
- With a rate in and a rate out, the net rate is in − out. The amount increases exactly when in > out.
Key relationships
| Idea | Formula | Units |
|---|---|---|
| Net change | ∫ (a to b) Q′(t) dt = Q(b) − Q(a) | rate units × time units |
| Amount function | Q(x) = Q(a) + ∫ (a to x) Q′(t) dt | units of Q |
| Rate of the amount | Q′(x) = rate at time x | units of Q per unit time |
| In and out | Q(x) = Q(a) + ∫ (a to x) [I(t) − O(t)] dt | units of Q |
| Total change | ∫ (a to b) |Q′(t)| dt | units of Q |
| Greatest/least amount | Compare Q at endpoints and where Q′ changes sign | units of Q |
Assumptions behind the method
- The rate is continuous on the interval, so the Fundamental Theorem applies.
- The rate and the input use matching units (do not mix minutes and hours).
- The starting value belongs to the same time as the lower limit of the integral.
Mistakes to avoid
- Leaving out the starting amount.
- Using rate(b) − rate(a) as the change in the amount.
- Saying the amount peaks when the rate peaks. It peaks where the net rate changes from + to −.
- Skipping the endpoints in the candidates test.
- Writing out − in instead of in − out.
- Rounding a critical value before using it in a later integral.
- Interpretations with no units or no time interval.
Quick self-check
- A tank holds 30 litres at t = 0. Water flows in at 4 + 2t litres per minute. How much is in the tank at t = 5? (30 + ∫ (0 to 5) (4 + 2t) dt = 30 + 45 = 75 litres)
- r(t) is in litres per minute and t in minutes. What are the units of ∫ (0 to 10) r(t) dt? (litres)
- Rate in > rate out at t = 2. Is the amount increasing or decreasing at t = 2? (Increasing, because the amount’s derivative is in − out > 0.)
Next: practice questions.