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Average Value of a Function

Quick Navigation

• Key Formula — The boxed formula you need to memorize
• Where This Comes From — Derivation from Riemann sums
• Worked Example — Complete solution walkthrough
• Practice Problems
• Levels 1-2 — Build confidence
• Levels 4-5 — Challenge problems
• Common Pitfalls — Mistakes to avoid
• Conceptual Questions — Test your understanding

Before You Start

Do you remember what a Riemann sum represents?

A Riemann sum $\sum_{i=1}^n f(x_i^*)\Delta x$ approximates the area under a curve by adding up rectangles.

From Discrete to Continuous Averaging

How do you find the "average temperature" over an entire day when temperature changes continuously? You can't just add up infinitely many values and divide. Yet averaging is one of the most natural things we do with data—so there must be a way to extend it to continuous functions.

The key insight: averaging finitely many values becomes an integral when we have infinitely many. This connection between sums and integrals is exactly what the definite integral was designed for.

Prerequisite Map

graph LR
    subgraph Foundation["Foundation (Earlier Chapters)"]
        AA["Limits"]
        AB["Antiderivatives"]
    end

    subgraph DirectPrereqs["Direct Prerequisites"]
        A["Definite Integrals<br/>(min level 3)"]
        B["FTC Part 2<br/>(min level 2)"]
        AC["Riemann Sums<br/>(conceptual)"]
    end

    subgraph ThisSkill["This Skill"]
        C["Average Value<br/>Formula"]
    end

    subgraph Unlocks["What This Unlocks"]
        D["Mean Value Theorem<br/>for Integrals"]
        E["Probability<br/>Density Functions"]
        F["Expected Value<br/>(Statistics)"]
    end

    AA --> AC
    AB --> B
    AC --> A
    A --> C
    B --> C
    AC -.->|"intuition"| C
    C --> D
    C --> E
    D --> F

    style C fill:#d1fae5,stroke:#a565f0,stroke-width:3px
    style A fill:#fef3c7,stroke:#f59e0b
    style B fill:#fef3c7,stroke:#f59e0b

    click A "../ch4-sec2/definite-integral-evaluation.html"
    click B "../ch4-sec3/ftc-part2.html"
    click AC "../ch4-sec2/riemann-sums.html"
    click D "mean-value-theorem-integrals.html"
    click E "../ch8-sec5/probability-density.html"

    click C "average-value-formula.html"
    click F "../ch8-sec5/expected-value.html"

Legend: Yellow nodes = direct prerequisites you need. Green = this skill. Dashed arrows = helpful but not required.

Quick Reference

Key Concepts

The Average Value Formula

For a continuous function $f$ on the interval $[a, b]$, the average value is:

$$\boxed{f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\,dx}$$

Breaking this down:

• $\int_a^b f(x)\,dx$ = total "accumulated" quantity
• $b - a$ = length of the interval
• $\frac{1}{b-a}$ = the "averaging" factor that spreads the total evenly

Where This Comes From

For $n$ discrete values $y_1, y_2, \ldots, y_n$, the average is:

$$y_{\text{avg}} = \frac{y_1 + y_2 + \cdots + y_n}{n}$$

For a function sampled at $n$ points with spacing $\Delta x = \frac{b-a}{n}$:

$$\frac{f(x_1^*) + f(x_2^*) + \cdots + f(x_n^*)}{n} = \frac{1}{b-a} \sum_{i=1}^n f(x_i^*)\Delta x$$

As $n \to \infty$, the Riemann sum becomes the integral:

$$f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\,dx$$

Visualization

    y
    │        ╭───╮
    │      ╱│     │╲
f_avg├─────┼─────────┼──────  ← horizontal line at average height
    │    ╱ │         │ ╲
    │   ╱  │         │  ╲
    │  ╱   │         │   ╲
    └──────┼─────────┼──────→ x
           a         b

    Area under curve = Area of rectangle with height f_avg

The Rectangle Interpretation

Key insight: 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.

$$\text{Area under curve} = \int_a^b f(x)\,dx = f_{\text{avg}} \cdot (b-a) = \text{Area of rectangle}$$

Physical Interpretation

The average value appears naturally in many contexts:

• Temperature: Average temperature over a day
• Velocity: Average speed over a trip (distance/time)
• Density: Average density of a non-uniform rod
• Power: Average power consumption over time
• Concentration: Average chemical concentration in a solution

Common Pitfalls

💡 Don't worry if you make these mistakes—almost everyone does at first!

Worked Example

Problem: Find the average value of $f(x) = x^2 - 2x + 3$ on the interval $[1, 4]$.

Solution:

Step 1: Identify the interval.

• $a = 1$, $b = 4$
• Interval length: $b - a = 4 - 1 = 3$

Step 2: Write down the formula with our values plugged in: $$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx = \frac{1}{3}\int_1^4 (x^2 - 2x + 3)\,dx$$

Step 3: Find the antiderivative: $$\int (x^2 - 2x + 3)\,dx = \frac{x^3}{3} - x^2 + 3x + C$$

Step 4: Evaluate at the bounds: $$\left[\frac{x^3}{3} - x^2 + 3x\right]_1^4 = \underbrace{\left(\frac{64}{3} - 16 + 12\right)}_{\text{at } x=4} - \underbrace{\left(\frac{1}{3} - 1 + 3\right)}_{\text{at } x=1}$$

$$= \left(\frac{64}{3} - 4\right) - \left(\frac{1}{3} + 2\right) = \frac{64}{3} - 4 - \frac{1}{3} - 2 = \frac{63}{3} - 6 = 21 - 6 = 15$$

Step 5: Apply the averaging factor: $$f_{\text{avg}} = \frac{1}{3} \cdot 15 = 5$$

Practice Problems

How to use these problems:

• Levels 1-2: Build confidence with the basic formula
• Level 3: Add complexity (trig, substitution)
• Levels 4-5: Applications and proofs—exam-level difficulty

CCI-Style Conceptual Questions

These questions test understanding, not computation. They're the type that separate students who memorized from those who truly understand.

Question 1: If $\int_0^{10} f(x)\,dx = 50$, what is the average value of $f$ on $[0, 10]$?

Question 2: The average value of a function $f$ on $[2, 8]$ is 7. What is $\int_2^8 f(x)\,dx$?

Question 3: Can the average value of a function on an interval be larger than the maximum value of the function on that interval? Explain.

Question 4: True or False: If two different functions have the same average value on $[a, b]$, they must have the same integral on $[a, b]$.

Question 5: A function $f$ is positive on $[0, 5]$ with $f_{\text{avg}} = 4$. Sketch a possible graph of $f$, and shade the rectangle with the same area as the region under $f$.

    y
    │     ╭──╮
  4 ├─────┼──┼─────  ← rectangle height
    │    ╱│  │╲
    │   ╱ │  │ ╲
    │  ╱  │  │  ╲
    └──────────────→ x
         0    5

Mastery Checklist

Level 1-2: Novice → Competent

• [ ] I can write the average value formula from memory: $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$
• [ ] I can compute the average value of polynomial functions on given intervals
• [ ] I can identify $a$, $b$, and compute $(b-a)$ correctly, including when $a < 0$

Level 3: Competent → Proficient

• [ ] I can compute average values involving trig functions (knowing their antiderivatives)
• [ ] I can interpret average value geometrically as the height of an equal-area rectangle
• [ ] I understand that $f_{\text{avg}}$ must lie between $\min(f)$ and $\max(f)$ on $[a,b]$

Level 4-5: Proficient → Expert

• [ ] I can apply average value to physics contexts (velocity, density, temperature)
• [ ] I can distinguish between average velocity and average speed
• [ ] I can prove inequalities involving average values using integral properties
• [ ] I can explain why the formula works (derivation from Riemann sums)

Self-Assessment

If you can check all boxes in Levels 1-3, you're ready for exams on this topic. Levels 4-5 prepare you for harder problems and future courses.

Mental Model

The "Smoothing" Analogy:

Imagine pouring water into a container shaped like the region under a curve. The average value is the water level if the container had straight vertical walls—all the peaks and valleys have been "averaged out" to a single uniform height.

Key Takeaways

📌 What to remember for exams:

1. The formula: $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$
2. Geometric meaning: Height of equal-area rectangle
3. Bounds: $\min(f) \leq f_{\text{avg}} \leq \max(f)$
4. Common trap: Don't forget to divide by $(b-a)$!

Connections

Looking back:

• Definite integrals provide the computational foundation
• This formula is just a Riemann sum interpretation applied differently

Looking ahead:

• Mean Value Theorem for Integrals guarantees the average is actually achieved
• Probability density functions use average value to define expected value

Real-world connections:

• In physics, average velocity = total displacement / time
• In statistics, expected value is the "average" of a continuous distribution

Last updated: 2026-01-23