AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/8/8-1)
Unit 8 · Topic 8.1
8.1 Finding the Average Value of a Function on an Interval
The average value of a function f on [a, b] is the integral of f over the interval divided by its length, b − a. It's the height of the rectangle that has the same area as the region under the curve, and it's different from the average rate of change.
Key terms
- average value
- definite integral
- interval length (b − a)
- net area
The formula
For a continuous function f on [a, b], the average value is
average value of f = (1/(b − a)) ∫ₐᵇ f(x) dx.
Think about averaging test scores: add them up and divide by how many there are. A function has infinitely many values, so the integral does the adding and the interval's length b − a does the dividing.
The picture
Rearrange the formula: (average value) × (b − a) = ∫ₐᵇ f(x) dx. So the average value is the height of a rectangle over [a, b] whose area equals the (signed) area under f. If you flattened the region under the curve into a rectangle on the same base, its height would be the average value. Parts of the curve stick up above that height and parts dip below, and the extra and missing areas balance out.
Because the integral counts area below the axis as negative, the average value can be negative or zero.
Average value vs. average rate of change
These sound alike but answer different questions. Mixing them up is one of the most common errors in the course.
| Question | Formula | Example with f(x) = x² on [1, 3] |
|---|---|---|
| Average value of f | (1/(b − a)) ∫ₐᵇ f(x) dx | (1/2)(26/3) = 13/3 |
| Average rate of change of f | (f(b) − f(a))/(b − a) | (9 − 1)/2 = 4 |
Average value in context
If v(t) is velocity, the average value of v over [a, b] is the average velocity, (1/(b − a)) ∫ₐᵇ v(t) dt, which equals displacement divided by time. The average acceleration is the average rate of change of v, (v(b) − v(a))/(b − a). Both are averages, but they use different formulas because they average different things.
When f is given only in a table, estimate the integral with a Riemann or trapezoidal sum, then divide by b − a.
One more fact: if f is continuous on [a, b], it actually reaches its average value somewhere. There's at least one c in [a, b] where f(c) equals the average value. This follows from the Intermediate Value Theorem, because the average lies between f's minimum and maximum on the interval.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Average value, and where f reaches it
Find the average value of f(x) = x² on [1, 3]. Then find the value c in [1, 3] where f(c) equals that average.
Show the solutionHide the solution
- Step 1: ∫₁³ x² dx = [x³/3]₁³ = 9 − 1/3 = 26/3.
- Step 2: Divide by b − a = 2: average value = (26/3)/2 = 13/3 ≈ 4.333.
- Step 3: Solve c² = 13/3: c = √(13/3) ≈ 2.082, which is in [1, 3]. (The negative root isn't in the interval.)
- Step 4: Compare: the average rate of change of f on [1, 3] is (9 − 1)/2 = 4, a different number.
Answer: Average value 13/3; c = √(13/3) ≈ 2.082
- Example 2
Average value from a table
The temperature T(t), in °F, is recorded at t = 0, 2, 6 and 8 hours: 50, 56, 64 and 60. Use a trapezoidal sum to estimate the average temperature over 0 ≤ t ≤ 8.
Show the solutionHide the solution
- Step 1: Trapezoidal sum for ∫₀⁸ T(t) dt: ½(50 + 56)(2) + ½(56 + 64)(4) + ½(64 + 60)(2) = 106 + 240 + 124 = 470.
- Step 2: Divide by the length of the interval, 8: 470/8 = 58.75.
- Step 3: Units: the integral is in °F·hours; dividing by hours gives °F.
Answer: About 58.75°F
Common mistakes
- Computing the average rate of change, (f(b) − f(a))/(b − a), when the question asks for average value.
- Forgetting to divide by b − a.
- Averaging the table values directly, like (50 + 56 + 64 + 60)/4. That ignores the uneven spacing; use a sum that weights by width.
- Dropping the units. The average value has the same units as f.
On the exam
- Free-response questions often ask for “the average value of R(t) over the interval,” sometimes from a table. Write (1/(b − a)) ∫ₐᵇ R(t) dt before computing.
- If a question asks for average velocity, use the average value of v. If it asks for average acceleration, use the average rate of change of v.
Connected topics
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Check yourself
4 questions on 8.1 Finding the Average Value of a Function on an Interval. Pick an answer to see if you got it, and why.
The function f is continuous on [0, 6], and the average value of f on [0, 6] is 4. What is the value of ∫₀⁶ (2f(x) + 1) dx?
The depth of the water in a tidal pool, in feet, is modeled by D(t) = 3 + ln(1 + t²), where t is measured in hours for 0 ≤ t ≤ 4. What is the average depth of the water over this time interval?
Let f(x) = √x. For what value of c in the interval [0, 4] is f(c) equal to the average value of f on [0, 4]?
The function f is continuous for all real numbers. The average value of f on [1, 4] is 5, and the average value of f on [1, 7] is 2. What is the average value of f on [4, 7]?
0 of 4 answered