AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/8/8-3)
Unit 8 · Topic 8.3
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
If you know the rate at which a quantity changes, the definite integral of that rate gives the total change, and adding the starting amount gives the final amount. This topic applies that idea to water, people, cars, money and other real contexts, with careful units.
Key terms
- rate of change
- net change
- accumulation function
- starting amount
- units
The core equation
If Q(t) is an amount and Q′(t) is its rate of change, then
Q(b) = Q(a) + ∫ₐᵇ Q′(t) dt.
The integral is the net change from a to b. If the problem gives a rate in “per hour,” the integral is in the same units as the amount (the hours cancel).
Rates in and rates out
Many problems give two rates: something entering at E(t) and leaving at L(t). The net rate is E(t) − L(t), and
amount at time b = amount at time a + ∫ₐᵇ [E(t) − L(t)] dt.
The amount is increasing when E(t) > L(t) and decreasing when E(t) < L(t). It reaches a local maximum when the net rate changes from positive to negative, that is, where E(t) = L(t) and E switches from bigger to smaller.
Accumulation functions in context
A function like A(x) = 50 + ∫₀ˣ r(t) dt gives the amount at time x. Its derivative is A′(x) = r(x), by the Fundamental Theorem. So questions like “Is the amount increasing at x = 5?” are really “Is r(5) positive?”, and “Find the maximum amount” is a Candidates Test (5.5) using the zeros of r and the endpoints.
Explaining the meaning of an integral
When asked what ∫ₐᵇ r(t) dt means, give a complete sentence with three parts: the quantity, the units and the time interval. Also say whether it's a total amount or a change, as appropriate.
For example, if P′(t) is the rate (in people per hour) at which people enter a park, then “∫₂⁵ P′(t) dt = 300 means 300 people entered the park from t = 2 to t = 5 hours.” If P′ can be negative (people leaving), say it's the net change in the number of people in the park.
Checklist
- Find the starting amount and the rate (or rates) in the problem.
- Write the integral setup with the correct limits.
- Use the calculator for the value if it's allowed and the integrand is messy.
- Add the starting amount if the question asks for an amount, not a change.
- Include units in the final answer.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Water in, water out (calculator)
A tank holds 150 gallons at t = 0. Water flows in at E(t) = 15 + 5 cos(t/3) gallons per hour and drains at L(t) = 2t + 6 gallons per hour, for 0 ≤ t ≤ 12. (a) How much water is in the tank at t = 12? (b) At what time is the amount greatest, and how much is there then? Justify.
Show the solutionHide the solution
- Step 1: (a) W(12) = 150 + ∫₀¹² [E(t) − L(t)] dt. With a calculator, ∫₀¹² E(t) dt ≈ 168.648 and ∫₀¹² L(t) dt = 216, so W(12) ≈ 150 + 168.648 − 216 ≈ 102.648 gallons.
- Step 2: (b) W′(t) = E(t) − L(t). Solve E(t) = L(t) with the calculator: t ≈ 4.597.
- Step 3: E − L > 0 for t < 4.597 (at t = 0, E − L = 20 − 6 = 14), and E − L < 0 after. So W increases, then decreases.
- Step 4: Candidates: W(0) = 150, W(4.597) = 150 + ∫ from 0 to 4.597 of (E − L) dt ≈ 185.229, W(12) ≈ 102.648.
- Step 5: The largest is at t ≈ 4.597.
Answer: (a) About 102.648 gallons. (b) About 185.229 gallons, at t ≈ 4.597 hours, where W′ = E − L changes from positive to negative.
- Example 2
Trap: interpreting with the right words
R(t) is the rate, in vehicles per minute, at which cars pass a toll booth, where t is in minutes. Explain the meaning of ∫₁₀³⁰ R(t) dt and of (1/20)∫₁₀³⁰ R(t) dt.
Show the solutionHide the solution
- Step 1: Units of the first: (vehicles per minute) × (minutes) = vehicles.
- Step 2: ∫₁₀³⁰ R(t) dt is the total number of vehicles that pass the booth from t = 10 to t = 30 minutes.
- Step 3: The second divides by the interval's length (30 − 10 = 20 minutes): it's the average rate, in vehicles per minute, over that interval (8.1).
- Step 4: Weak answers say “the area under R” or “the number of cars at t = 30.” The integral is a count over the interval, not an amount at an instant.
Answer: ∫₁₀³⁰ R(t) dt = the number of vehicles that pass between t = 10 and t = 30 minutes; dividing by 20 gives the average rate, in vehicles per minute, over that interval.
Common mistakes
- Forgetting the starting amount when asked “how much is in the tank.”
- Integrating E(t) and L(t) correctly but subtracting in the wrong order.
- Finding where E = L and calling it the maximum without checking the endpoints or the sign change.
- Interpreting an integral of a rate as a rate, or as an amount at one instant.
On the exam
- This is the classic calculator free-response question: a rate in, a rate out, a starting amount. Common parts are total change, amount at a time, when the amount is greatest, and meaning of an expression.
- Store intermediate values in your calculator rather than rounding them, then round the final answer to three decimals.
Connected topics
Videos
Check yourself
5 questions on 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts. Pick an answer to see if you got it, and why.
Rainwater collects in a barrel at the rate R(t) = 3√t · e^(−0.2t) liters per hour for 0 ≤ t ≤ 8, where t is measured in hours. At the same time, water leaks out of the barrel at a constant rate of 1.2 liters per hour. At time t = 0 the barrel holds 12 liters. How many liters of water are in the barrel at time t = 8?
An amusement park opens at 9 a.m. with no visitors inside. For 0 ≤ t ≤ 10, where t is the number of hours after 9 a.m., people enter the park at a rate modeled by E(t) = 800 + 400 sin(t/2) people per hour and leave the park at a rate modeled by L(t) = 150t people per hour.
Invented model of park attendance
At what time t, for 0 ≤ t ≤ 10, is the number of people in the park greatest?
To the nearest whole number, how many people are in the park at the time when the number of people in the park is greatest?
| t (hours) | W(t) (gallons per hour) |
|---|---|
| 0 | 12 |
| 2 | 15 |
| 5 | 11 |
| 6 | 9 |
| 10 | 4 |
Invented data: rate of water flowing into a tank
Water flows into a tank at the rate W(t) gallons per hour, where t is measured in hours. Selected values of W(t) are given in the table. At time t = 0 the tank holds 40 gallons. Using a trapezoidal sum with the four subintervals shown in the table, what is the best estimate of the amount of water in the tank at time t = 10?
W(t) is the rate, in gallons per hour, at which water flows into the tank at time t hours. Which of the following is the best interpretation of (1/10) ∫ from 0 to 10 of W(t) dt?
0 of 5 answered