Topic 8.1 Finding the Average Value of a Function on an Interval: compute the average value of a function with the definite-integral formula.

What this topic is asking

The College Board (Topic 8.1) gives the average value of a continuous function over an interval, computed with a definite integral. It is the integral analogue of averaging a list of numbers, and it must not be confused with the average rate of change.

The formula

A worked average value

Average value versus average rate of change

The exam deliberately tests the distinction between two "average" quantities. The average value of $f$ uses an integral, $\frac{1}{b-a}\int_a^b f$, and answers "what constant height matches the area under $f$". The average rate of change of $f$ uses endpoint values, $\frac{f(b) - f(a)}{b-a}$, and answers "what was the average slope". They are different computations for different questions: average value needs $\int f$, average rate needs $f$ at the endpoints. A clue is the wording: "average value of the velocity" wants the integral of velocity over the width; "average rate of change of position" wants the displacement over the width (which is the average velocity, equal to the average value of velocity by the Fundamental Theorem).

The Mean Value Theorem for integrals

For a continuous function, the Mean Value Theorem for integrals guarantees at least one point $c$ in $[a, b]$ where $f(c) = f_{\text{avg}}$: the function actually attains its average value somewhere on the interval. This justifies the geometric picture of an equal-area rectangle whose height is reached by the curve. On the exam you may be asked to find the time or place where the function equals its average value, which you do by setting $f(x) = f_{\text{avg}}$ and solving, as in the worked free-response above. The most common error in this whole topic is forgetting the $\frac{1}{b-a}$ factor and reporting the integral itself as the average value.

Average value in applied contexts

Average value appears most often dressed as a real quantity: the average temperature over a day, the average velocity over a trip, the average concentration over an interval. In each case you integrate the quantity's function over the interval and divide by the interval's width, then report the result with the quantity's units. A subtle point is that the average velocity computed this way, $\frac{1}{b-a}\int_a^b v\,dt$, equals the displacement divided by the elapsed time, because $\int v\,dt$ is the displacement; this matches the everyday notion of average velocity and connects the integral formula to the familiar distance-over-time idea. Interpreting the average value in the problem's own terms, with units, is what the free-response section expects.

On the calculator-active section

When the function is complicated, average-value problems live on the calculator-active part, where you evaluate $\int_a^b f$ numerically and then divide by $b - a$. The setup is still the load-bearing step: write $\frac{1}{b-a}\int_a^b f(x)\,dx$ explicitly with the correct limits before computing. Present the numerical answer to three decimal places when the value is not exact, and include units. The same discipline applies to finding where the function equals its average value, you may need the calculator to solve $f(x) = f_{\text{avg}}$ numerically. Keeping the exact setup visible ensures you earn the method marks even if the arithmetic is done by the calculator.