Average Value of a Function

What Does "Average Value" Mean?

For a list of numbers, finding an average is straightforward: add the values and divide by how many values there are. But what happens when you want the average value of a function over an entire interval?

For a continuous function, there are infinitely many values between and . Instead of adding individual values, calculus uses a definite integral to accumulate the function's values over the interval.

The average value is therefore the total accumulated value divided by the length of the interval.

The Average Value Formula

If is continuous on the interval , its average value is:

This formula has a simple interpretation:

• measures the total accumulated value of the function over the interval.
• is the length of the interval.
• Dividing the accumulated value by the interval length gives the function's average value.

Why the Formula Makes Sense

Think about a constant function first. Suppose on an interval of length . Its average value should obviously be .

Using the formula:

The integral gives the area of a rectangle: height times width. Dividing by the width leaves the height. So the formula correctly returns the constant value.

For a nonconstant function, the same idea applies. The integral gives the total "amount" accumulated by the function, and dividing by the interval length gives the height of a constant function that would have the same total accumulated value.

A Step-by-Step Method

1. Identify the function and interval. Find and the endpoints and .
2. Write the average value formula. Use .
3. Evaluate the definite integral. Use an antiderivative, the Fundamental Theorem of Calculus, or another appropriate method.
4. Divide by the interval length. Remember that the denominator is , not simply one of the endpoints.
5. Interpret the result if necessary. The answer represents the average height/value of the function over the interval.

Worked Examples

Average Value vs. Average Rate of Change

These two ideas sound similar, but they are completely different.

• Average value: asks for the average height of the function over an interval.
• Average rate of change: measures how much the function changes per unit of input.

The average value of on is:

The average rate of change over the same interval is:

Do not confuse these formulas on the AP exam. The average value formula contains a definite integral; the average rate of change formula does not.

The Mean Value Theorem for Integrals

The average value formula is closely connected to the Mean Value Theorem for Integrals.

If is continuous on , then there is at least one number in the interval such that:

Dividing both sides by gives:

That means that for a continuous function, there is at least one point in the interval where the function's actual value equals its average value.

This is an important conceptual connection: the average value isn't merely a number calculated from an integral—it is actually attained somewhere on the interval.

Common Mistakes

Practice Problems