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Average Value of
Functions on Intervals

Master the technique of finding the average height of a function over a closed interval using definite integration.

AP Calculus BC

What Is the Average Value of a Function?

You already know how to average a list of numbers: add them up and divide by the count. But what happens when you have infinitely many values along a continuous curve? The average value of a function extends this idea by replacing the sum with an integral and the count with the length of the interval.

Visual Intuition: Area Redistribution

x y a b \(f_{\text{avg}}\) \(f(x)\) Same total area

Why Does Average Value Matter?

Physics
Average velocity, average force, and average temperature over time intervals.
Economics
Average cost or revenue over a production range.
AP Exam
Frequently tested on both AP Calculus AB and BC: free-response and multiple choice.

The Average Value Formula

If \(f\) is continuous on \([a,b]\), its average value is:

1
\(\int_a^b f(x)\,dx\): the total "accumulated value" (area with sign) under the curve.
2
\(\frac{1}{b-a}\): divides by the interval length, just like dividing a sum by the count.

Mean Value Theorem for Integrals

The average value formula is directly tied to the Mean Value Theorem for Integrals. It guarantees that there exists at least one point \(c\) in \([a,b]\) where the function actually equals its average value:

Finding \(c\) on the Graph

x y a b \(f(x)\) \(f_{\text{avg}}\) \(c_1\) \(c_2\) \(f(c_1) = f(c_2) = f_{\text{avg}}\)

Average Value vs. Average Rate of Change

Average Value Average Rate of Change
Formula \(\frac{1}{b-a}\int_a^b f(x)\,dx\) \(\frac{f(b)-f(a)}{b-a}\)
Measures Average height of the curve Average slope (secant line)
Requires Integration Only endpoint values
Related MVT MVT for Integrals MVT for Derivatives

Step-by-Step Process

1
Identify the interval \([a,b]\) and the function \(f(x)\).
2
Set up the formula: \(f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx\).
3
Evaluate the definite integral using antidifferentiation (or the Fundamental Theorem of Calculus).
4
Multiply the integral result by \(\frac{1}{b-a}\) to get the average value.
5
If asked for \(c\): set \(f(c) = f_{\text{avg}}\) and solve for \(c \in [a,b]\).

Example 1: \(f(x) = x^2\) on \([0,3]\)

x y 0 1 2 3 3 6 9 \(f(x) = x^2\) \(f_{\text{avg}} = 3\)

Example 1 Continued: Finding \(c\)

We found \(f_{\text{avg}} = 3\). Now find the value \(c \in [0, 3]\) where \(f(c) = 3\):

Example 2: \(f(x) = \sin x\) on \([0, \pi]\)

x y 0 \(\frac{\pi}{2}\) \(\pi\) 1 \(f(x) = \sin x\) \(\frac{2}{\pi}\)

Example 3: \(f(x) = e^{2x}\) on \([0, 1]\)

This example requires a u-substitution when integrating the exponential function.

Exact Value
\(\frac{e^2 - 1}{2}\)
Approximate
\(\approx 3.195\)

Signed Area and Negative Values

x \(\sin x\) + area − area 0 \(\pi\) \(2\pi\)

Common Mistakes to Avoid

✕ Forgetting \(\frac{1}{b-a}\)
Computing \(\int_a^b f(x)\,dx\) alone gives the total area, not the average. You must divide by the interval length.
✕ Wrong MVT
Confusing the MVT for Integrals with the MVT for Derivatives. Average value uses integrals; average rate of change uses endpoints.
✕ Ignoring Sign
The integral gives signed area. If the function dips below the x-axis, those regions subtract from the total. The average value can be negative.

Summary: Key Formulas

Concept Formula Key Idea
Average Value \(\frac{1}{b-a}\int_a^b f(x)\,dx\) Height of equal-area rectangle
MVT for Integrals \(f(c) = f_{\text{avg}}\) for some \(c\) Function hits its average
Avg Rate of Change \(\frac{f(b)-f(a)}{b-a}\) Secant line slope
Total Accumulation \(\int_a^b f(x)\,dx\) Signed area under curve

Question 1 of 5

What is the average value of \(f(x) = 3x^2\) on the interval \([1, 3]\)?

Question 2 of 5

The average value of \(f(x) = \cos x\) on \([0, \frac{\pi}{2}]\) is:

Question 3 of 5

If the average value of \(f\) on \([2, 6]\) is \(5\), what is \(\int_2^6 f(x)\,dx\)?

Question 4 of 5

The average value of \(f(x) = x^2\) on \([0,3]\) is \(3\). Find the value of \(c\) in \([0,3]\) guaranteed by the MVT for Integrals.

Question 5 of 5

What is the average value of \(f(x) = \sin x\) on the interval \([0, 2\pi]\)?