Exploring Accumulations of Change
Accumulate signed rate contributions and distinguish net change from total activity and final amount.
College Board Topic 6.1 · CED p. 113 (PDF p. 118) · CED effective Fall 2020; project course outline 2026–27.
A rate tells you how fast a quantity is changing. Multiply rate by a short time interval and you get a small change; add those changes to get the whole story. On a rate graph, that sum becomes signed area. Above the axis adds to the quantity; below the axis subtracts. Net change and total activity answer different questions.
Keep units attached: (liters/minute) × minutes gives liters. An integral of a rate gives change, so an initial amount must be added if you want a final amount. Geometry works when the graph really consists of familiar shapes; a curved region is not automatically a triangle.
Let's work through it
- The positive triangle contributes ½(2)(3) = 3 units. Reason: Its rate is positive and triangle area is half base times height.
- The negative triangle contributes −½(2)(2) = −2 units. Reason: Signed area below the axis is negative.
- Net change is 3 − 2 = 1 unit; total absolute change is 3 + 2 = 5 units. Reason: Net change retains signs; total activity counts both directions.
Try it from memory
Pause & retrieve
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 flow rate is in gallons/hour and time is in hours. What are the integral’s units?
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 flow rate is in gallons/hour and time is in hours. What are the integral’s units?
Worked answer and reason
Gallons.
Multiply rate units by time units.
Retrieval 2
A rate stays below zero. Is accumulated change positive or negative?
Worked answer and reason
Negative.
Every small signed contribution is negative.
Catch the tempting mistake
- An integral always measures ordinary positive area.
- A definite integral gives signed accumulation. Use absolute value or split at zeros when the question asks for total activity.
Make the picture move
Explore → predict → explain
Below the axis counts with a minus sign.
Try first. Predict whether the signed integral and the total geometric area are equal. Then change the sampling rule and the number of rectangles.
Scroll sideways to see the full graph on a small screen. The table below gives the numerical values.
- Function: x − 1
- Solid rectangles: positive · dashed rectangles: negative
midpoint sum with 8 rectangles: 1.5. Exact signed integral: 1.5. Total geometric area: 2.5.
Compare the numbers
| Rectangles n | midpoint sum | Exact signed integral | Absolute error |
|---|---|---|---|
| 8 | 1.5 | 1.5 | 0 |
| 16 | 1.5 | 1.5 | 0 |
| 32 | 1.5 | 1.5 | 0 |
What’s going on? The negative triangle has area 1/2; the positive triangle has area 2. Their signed total is −1/2 + 2 = 3/2. Their geometric areas add to 5/2. The midpoint rule is exact for this linear function.
Keep in mind. The displayed rectangles approximate signed accumulation. They do not automatically approximate total area: that requires summing |f|. More rectangles need not strictly improve every example or every sampling rule.
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
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 1 · No calculator · graphical
- Unit 6 problem 2 · No calculator · verbal
For the teacher
- Ask students to tell a story about the graph before calculating. Require the rate, interval, sign, and units in a one-sentence interpretation.
Glossary and deeper reading · 3 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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Net change
Net change is the final amount minus the initial amount. Integrating a rate gives this signed change when the model and integral are valid. For example, moving 5 meters right and then 2 meters left gives net displacement 3 meters. Opposite contributions cancel, so net change can differ from the total amount of movement or activity.
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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.