Finding the Average Value of a Function on an Interval
Use an integral average and its units, rather than a secant rate or an unweighted mean of unequal-time samples.
College Board Topic 8.1 · CED p. 147 (PDF p. 152) · CED effective Fall 2020; project course outline 2026–27.
For continuous f on [a,b], its average value is (1/(b−a))∫ₐᵇ f(x)dx. Picture a constant-height rectangle whose signed area matches the integral: height times width equals total accumulation. The average value has the same units as f. It is different from average rate of change, which uses [f(b)−f(a)]/(b−a).
For unevenly spaced samples, approximate the integral with interval widths first, then divide by the total interval length. Simply averaging all listed heights weights every sample equally, which may not match elapsed time. If a question asks where the function takes its average value, solve f(c)=favg and give a c in the interval; continuity guarantees at least one such value.
Let's work through it
- For f(x)=x² on [0,3], compute ∫₀³ x²dx=9. Reason: Integrate the function, not its derivative.
- Divide by the interval length 3 to get favg=3. Reason: Average height is accumulated signed area per unit width.
- The value occurs at c=√3 in [0,3]. Reason: Solve c²=3 and retain a point in the interval.
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
What is the average value of a constant f=7?
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
What is the average value of a constant f=7?
Worked answer and reason
7.
Its integral is 7(b−a), and the interval length cancels.
Retrieval 2
What units does an average temperature have?
Worked answer and reason
Temperature units, such as °C.
Integration adds time units and division by interval length removes them.
Catch the tempting mistake
- Average value is the slope of the secant line.
- That is average rate of change. Average value is the integral divided by interval length.
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
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Put it to work
- Unit 8 problem 1 · No calculator · analytic
- Unit 8 problem 2 · No calculator · numerical
For the teacher
- Compare both averages for one function. Use nonuniform time data to expose why the plain mean of sample heights can be misleading.
Glossary and deeper reading · 4 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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Average value of a function
The average value of an integrable function on [a,b], with a<b, is its integral divided by b−a. It is the height of a constant function with the same signed accumulation over that interval. For example, x² on [0,3] has average value 9/3=3. This is an average of outputs, whereas an average rate of change compares endpoint outputs with input change.
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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.
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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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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.