Section 5.5: Average Value of a Function
| Course: MATH162 | Textbook: Stewart Calculus 9th Edition, Section 5.5 |
The Big Picture
You know how to average a list of numbers: add them up and divide by how many there are. But what if you have infinitely many values—like the temperature at every instant during a day?
This section answers that question: the average value of a continuous function is its integral divided by the interval length. This simple formula connects discrete averages to continuous ones through a limiting process—and has a beautiful geometric interpretation.
Key Equations
| Formula | Name | What It Computes |
|---|---|---|
| $f_{\text{avg}} = \displaystyle\frac{1}{b-a}\int_a^b f(x)\,dx$ | Average Value Formula | The “typical” value of $f$ on $[a,b]$ |
| $\int_a^b f(x)\,dx = f(c)(b-a)$ | Mean Value Theorem for Integrals | There exists $c \in [a,b]$ where $f(c) = f_{\text{avg}}$ |
Geometric Interpretation
The average value $f_{\text{avg}}$ is the height of a rectangle with base $[a,b]$ that has the same area as the region under the curve.
f(x)
│ ╭───────╮
│ ╱ ╲
f_avg ───┼──────────────── (rectangle height)
│ ╱ ╲
└─────────────────→ x
a b
The areas match: $\int_a^b f(x)\,dx = f_{\text{avg}} \cdot (b-a)$
Why This Formula?
From Discrete to Continuous
For $n$ equally-spaced sample points:
\[\text{Average} = \frac{f(x_1) + f(x_2) + \cdots + f(x_n)}{n}\]With $\Delta x = \frac{b-a}{n}$, we can rewrite this as:
\[\text{Average} = \frac{1}{b-a} \sum_{i=1}^{n} f(x_i) \Delta x\]As $n \to \infty$, the sum becomes an integral:
\[f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\,dx\]The formula isn’t arbitrary—it’s the natural limit of sampling more and more values.
Skills in This Section
| Skill | Description | Difficulty |
|---|---|---|
| Average Value of a Function | Applications of Integration | Beginner |
| Mean Value Theorem for Integrals | Applications of Integration | Intermediate |
Learning Path
| Skill | What You’ll Learn | Key Takeaway |
|---|---|---|
| Average Value Formula | $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$ | Divide total integral by interval length |
| Mean Value Theorem for Integrals | $f(c) = f_{\text{avg}}$ for some $c$ | Continuous functions achieve their average |
| Applications | Temperature, velocity, density | Average of a rate gives total change ÷ interval |
Exercise Coverage Map
| Exercises | Topic | Key Challenge |
|---|---|---|
| 1–8 | Basic average value | Compute integral, divide by $(b-a)$ |
| 9–12 | Find $c$ where $f(c) = f_{\text{avg}}$ | Apply Mean Value Theorem |
| 13–14 | Conceptual reasoning | Use MVT to prove existence |
| 15–16 | Graphical estimation | Estimate average from graph |
| 17–18 | Temperature models | Interpret average in context |
| 19–21 | Applied problems | Density, velocity, biology |
| 22–26 | Proofs and theory | Derive properties of average value |
Self-Assessment Quiz
Q1: What is the average value of $f(x) = x^2$ on $[0, 3]$?
\[f_{\text{avg}} = \frac{1}{3-0}\int_0^3 x^2\,dx = \frac{1}{3} \cdot \frac{x^3}{3}\Big\vert _0^3 = \frac{1}{3} \cdot 9 = 3\]
Q2: The average value of $f$ on $[1, 5]$ is 7. What is $\int_1^5 f(x)\,dx$?
Rearranging the formula:
\[\int_1^5 f(x)\,dx = f_{\text{avg}} \cdot (b-a) = 7 \cdot 4 = 28\]
Q3: If $\int_2^6 f(x)\,dx = 20$, what is the average value of $f$ on $[2, 6]$?
\[f_{\text{avg}} = \frac{20}{6-2} = \frac{20}{4} = 5\]
Q4: True or false: For any continuous $f$ on $[a, b]$, there is some point $c$ where $f(c) = f_{\text{avg}}$.
True. This is the Mean Value Theorem for Integrals.
Since $f$ is continuous and $f_{\text{avg}}$ lies between the minimum and maximum values of $f$ on $[a,b]$, the Intermediate Value Theorem guarantees that $f(c) = f_{\text{avg}}$ for some $c \in [a,b]$.
Q5: The temperature during a 12-hour period is $T(t) = 50 + 14\sin(\pi t/12)$ °F. What’s the average temperature?
\[T_{\text{avg}} = \frac{1}{12}\int_0^{12} \left(50 + 14\sin\frac{\pi t}{12}\right)\,dt\]
\[= \frac{1}{12}\left[50t - 14 \cdot \frac{12}{\pi}\cos\frac{\pi t}{12}\right]_0^{12}\]
\[= \frac{1}{12}\left[(600 + \frac{168}{\pi}) - (0 - \frac{168}{\pi})\right] = \frac{1}{12}\left(600 + \frac{336}{\pi}\right)\]
\[= 50 + \frac{28}{\pi} \approx 58.9°\text{F}\]
Deep Connections
Average Value and Average Velocity
For a position function $s(t)$, the average velocity over $[t_1, t_2]$ is:
\[v_{\text{avg}} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} = \frac{\Delta s}{\Delta t}\]But also:
\[v_{\text{avg}} = \frac{1}{t_2 - t_1}\int_{t_1}^{t_2} v(t)\,dt\]These are the same! By the Fundamental Theorem:
\[\int_{t_1}^{t_2} v(t)\,dt = s(t_2) - s(t_1)\]This confirms that the average of the velocity function equals the average velocity from physics.
Weighted Averages
The average value formula generalizes to weighted averages:
\[\bar{f} = \frac{\int_a^b f(x) w(x)\,dx}{\int_a^b w(x)\,dx}\]where $w(x)$ is a weight function. This appears in probability (expected value) and physics (center of mass).
Connection to the Fundamental Theorem
The Mean Value Theorem for Integrals is closely related to the Mean Value Theorem for derivatives. If $F’(x) = f(x)$, then:
- MVT for derivatives: $F’(c) = \frac{F(b) - F(a)}{b-a}$ for some $c$
- MVT for integrals: $f(c) = \frac{1}{b-a}\int_a^b f(x)\,dx$ for some $c$
These are essentially the same statement, since $F(b) - F(a) = \int_a^b f(x)\,dx$.
Common Mistakes to Avoid
| Mistake | Why It’s Wrong | Correct Approach |
|---|---|---|
| Forgetting to divide by $(b-a)$ | Get total instead of average | Average = integral ÷ interval length |
| Wrong interval length | $(b-a)$, not $(a-b)$ or just $b$ | Subtract: endpoint minus start |
| Confusing average value with average rate | Rate needs division by time; value doesn’t | Read carefully: is it asking for average of a function or average rate of change? |
| Thinking average must occur at midpoint | $c$ is usually NOT at $(a+b)/2$ | Average value occurs at some $c$, but location depends on $f$ |
Key Mathematical Themes
| Theme | How It Appears | Why It Matters |
|---|---|---|
| Limiting process | Average of $n$ samples → integral as $n \to \infty$ | Connects discrete to continuous |
| Existence theorem | MVT guarantees $f(c) = f_{\text{avg}}$ exists | Continuous functions behave predictably |
| Geometric interpretation | Rectangle with same area | Visualizes the formula |
| Integral as total | Integral ÷ length = average | Reinforces integral as accumulation |
Looking Ahead
The average value formula appears throughout applied mathematics:
- Statistics: Expected value $E[X] = \int x \cdot f(x)\,dx$ is a weighted average
- Physics: Center of mass is an average position weighted by density
- Signal processing: Moving averages smooth noisy data
- Economics: Average cost, average revenue use the same structure
This section completes Chapter 5’s exploration of applications of integration. The “slice, sum, take limits” approach has given us tools for area, volume, work, and average value—a remarkably versatile framework.
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|---|---|---|
| Section 5.4 | Chapter 5 | Chapter 6 |