Using Accumulation Functions and Definite Integrals in Applied Contexts
Write initial amount plus accumulated net rate, then justify inventory values, extrema, and targets in context.
College Board Topic 8.3 · CED p. 149 (PDF p. 154) · CED effective Fall 2020; project course outline 2026–27.
The net-change principle is Q(b)−Q(a)=∫ₐᵇ Q′(t)dt. Read a context carefully: a rate of arrivals alone measures added arrivals; a net inventory rate may already include departures. If inflow and outflow are given separately, Q(t)=Q(a)+∫ₐᵗ[I(u)−O(u)]du. State units and what the resulting number measures.
To find when an accumulated amount is largest on a closed interval, inspect where its net rate changes sign, then compare endpoint and critical-point values. To find when it reaches a target, solve the accumulation equation, checking the permitted interval and any physical restrictions. Calculator integration and root finding can help, but first write the equation and explain what each term means. A negative net rate can reduce a positive amount; it does not automatically mean the amount itself is negative.
Let's work through it
- A tank starts with 10 L and has net rate 6−2t L/min on [0,4]. Set V(t)=10+∫₀ᵗ(6−2u)du. Reason: Initial amount plus net accumulated rate gives inventory.
- V(t)=10+6t−t², with V′=6−2t=0 at t=3. Reason: The Fundamental Theorem and antiderivative give the model and critical point.
- V(0)=10, V(3)=19, V(4)=18, so the maximum is 19 L at 3 min. Reason: Compare candidates on the closed interval.
Try it from memory
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Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
A machine’s rate is items/minute. What does its rate integral count?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
A machine’s rate is items/minute. What does its rate integral count?
Worked answer and reason
Items produced over the interval.
Multiply rate units by time units.
Retrieval 2
If Q(a)=50 and ∫ₐᵇ Q′=−8, what is Q(b)?
Worked answer and reason
42.
Add the net change to the initial amount.
Catch the tempting mistake
- The integral of an inflow rate automatically gives the remaining inventory.
- Include the initial inventory and subtract any outflow rate unless the given rate is already net.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 8 problem 6 · No calculator · verbal
- Unit 8 problem 7 · No calculator · verbal
- Unit 8 problem 8 · No calculator · verbal
For the teacher
- Use a two-rate context with an explicitly labeled net rate. Require a sentence interpreting any calculator result with units and interval.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.