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Average Value of
Functions on Intervals

Learn how to find the average height of a function over a closed interval using definite integrals: the bridge between integration and everyday averages.

AP Calculus AB

What Does "Average" Mean?

In everyday math, the average of a finite set of numbers is their sum divided by how many there are:

But what if you have infinitely many values: every output of a continuous function \( f(x) \) across an interval? You cannot simply add them up. That is where calculus comes in.

From Discrete to Continuous

Discrete Average
Sum a finite list and divide by the count \( n \). Works for data points and test scores.
Continuous Average
Replace the sum with an integral and the count with the interval length \( b - a \).

Visual Intuition

x y a b \(f_{\text{avg}}\) Same Area ↓

The Average Value Formula

\( \displaystyle\int_a^b f(x)\,dx \)
The total accumulated value (area with sign) under the curve from \( a \) to \( b \).
\( \dfrac{1}{b - a} \)
Divides by the interval length, just like dividing by \( n \) in a discrete average.

Where Does the Formula Come From?

1
Sample \( n \) equally spaced points in \([a,b]\). Each sub-interval has width \( \Delta x = \frac{b-a}{n} \).
2
Discrete average: \( \displaystyle\frac{1}{n}\sum_{i=1}^{n} f(x_i) \).
3
Rewrite using \( n = \frac{b-a}{\Delta x} \):   \( \displaystyle\frac{1}{b-a}\sum_{i=1}^{n} f(x_i)\,\Delta x \).
4
Take the limit as \( n \to \infty \): the Riemann sum becomes \( \displaystyle\frac{1}{b-a}\int_a^b f(x)\,dx \).

Average Value vs. Average Rate of Change

Average Value Average Rate of Change
Question What is the mean height of \( f \)? How fast does \( f \) change overall?
Formula \( \dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx \) \( \dfrac{f(b) - f(a)}{b - a} \)
Uses Integration Only endpoint values
Result A y-value (output) A slope

Step-by-Step Method

1
Identify \( a \), \( b \), and \( f(x) \) from the problem statement.
2
Compute the definite integral \( \displaystyle\int_a^b f(x)\,dx \) using antiderivatives or given information.
3
Multiply by the prefactor \( \dfrac{1}{b-a} \).
4
Interpret: the result is the constant \( y \)-value whose rectangle has the same area as the region under the curve.

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

1 2 3 0 9 \(y=3\)

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

0 π/2 π 1 \(\tfrac{2}{\pi}\)

That is approximately 0.637: the dashed line sits at about 63.7% of the peak.

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

0 1 2 1 \(\approx 3.19\)

Mean Value Theorem for Integrals

This guarantees that the function actually attains its average value at some point \( c \) in the interval. The average is not just an abstract number: there is a real input where the function equals it.

Example 1
For \( x^2 \) on \([0,3]\): the average is 3, and \( x^2 = 3 \) at \( x = \sqrt{3} \approx 1.73 \).
Example 2
For \( \sin x \) on \([0,\pi]\): the average is \( \frac{2}{\pi} \), attained at \( x = \arcsin\!\left(\frac{2}{\pi}\right) \).

Watch Out: Signed Area

0 π 2π x + area - area

What If \( f \) Is Given as a Table?

On the AP exam, you may not have a formula. If \( f \) is given at discrete points, approximate the integral using a Riemann sum, trapezoidal rule, or other technique, then apply the average value formula.

1
Approximate \( \displaystyle\int_a^b f(x)\,dx \) with a trapezoidal sum or left/right Riemann sum.
2
Divide the result by \( b - a \) to get \( f_{\text{avg}} \).
3
State that the answer is an approximation (the integral was approximated).

Common Mistakes

✕ Forgetting the prefactor
Computing \( \int_a^b f(x)\,dx \) alone gives total area, not the average. You must divide by \( b - a \).
✕ Wrong interval length
Using \( b + a \) or just \( b \) instead of \( b - a \). Double-check endpoints.
✕ Confusing with AROC
Average Rate of Change uses \( \frac{f(b)-f(a)}{b-a} \). Average Value uses the integral formula: they are different concepts.

Quick Reference Summary

Concept Formula Key Idea
Average Value \( \frac{1}{b-a}\int_a^b f(x)\,dx \) Mean height of the curve
MVT for Integrals \( f(c) = f_{\text{avg}} \) for some \( c \) Function hits its average
Average Rate of Change \( \frac{f(b)-f(a)}{b-a} \) Slope of secant line
Total Change (Net) \( \int_a^b f'(x)\,dx = f(b)-f(a) \) FTC connects these

Question 1 of 5

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

Question 2 of 5

If \( \displaystyle\int_2^6 f(x)\,dx = 20 \), what is the average value of \( f \) on \([2, 6]\)?

Question 3 of 5

The average value of \( \cos x \) on \([0, \pi]\) is:

Question 4 of 5

By the Mean Value Theorem for Integrals, if \( f \) is continuous on \([a,b]\), then there exists \( c \in [a,b] \) such that:

Question 5 of 5

The average value of \( f(x) = 6x \) on \([0, 4]\) is 12. At what value of \( c \) does \( f(c) = 12 \)?