Average value of a function
Find the constant height with the same signed area on an interval.
Average value of a function is one of 7 applications formulas in the calculus section of this library, and it is used at ap · university level.
Why average value of a function works
The integral gives a total, and dividing a total by the width of the interval converts it into an amount per unit of input, exactly as dividing a sum by how many numbers there are averages a list. Geometrically it is the height of the rectangle over the interval with the same signed area as the curve.
What each symbol means
$[a,b]$ is the interval.
Average value of a function: when it holds
$a\ne b$ and $f$ must be integrable on the interval.
When it stops applying
The formula divides by the interval width, so it says nothing about a single point where the two ends coincide. It also assumes the function is integrable throughout, and an infinite spike inside the interval leaves the average undefined rather than merely large.
Average value of a function: a worked example
For $f(x)=x^2$ on $[0,3]$, $f_{avg}=\frac13[x^3/3]_0^3=3$.
The mistake to avoid
What people do: Averaging just the two endpoint values.
Why it goes wrong: That throws away everything the function does in between; for x squared on the interval from 0 to 3 it returns 4.5 when the true average is 3.
Do this instead: Integrate across the whole interval and then divide by its width.
Average value of a function: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $[a,b]$ is the interval.
- Check the conditions before substituting. $a\ne b$ and $f$ must be integrable on the interval.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Applications
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where average value of a function comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about average value of a function
Does the function ever actually reach its average value?
For a continuous function, yes. The mean value theorem for integrals guarantees at least one input inside the interval where the height equals the average exactly.
Why divide by the width and not by a count of points?
There are infinitely many points, so counting them is hopeless. The width of the interval takes over the role that the number of entries plays for a finite list.
Can an average value be negative?
Yes, whenever the signed area is negative. A function that spends most of the interval below the axis has a negative average even if it climbs high at one end.
Is this the average of the slope as well?
No. This averages the height of the function; averaging its slope instead would give the total change divided by the width, which is a different quantity.