# Average value

The average value of f on [a, b] is the integral of f over [a, b], divided by the length b - a.

Maths (university) · Part 05: Integration · Card 9 of 10 · Last updated 2026-10-19

## Analogy

Levelling out a bumpy water surface until it is flat: the flat height is the average.

## Example

**Average of x^2 on [0, 3]**

Integral from 0 to 3 is 9.
Length of interval is 3.
Average = 9 / 3 = 3.
It lies between 0 and 9, as it should.

## In practice

Gives mean temperature over a day, or mean speed over a trip, from a continuous formula.

## Practice

### Flashcards

- **What is the average value of f on [a, b]?** The integral of f from a to b, divided by b - a.

## Questions people ask

### Why divide by b minus a?

The integral is a total, like total distance. Dividing a total by the length of the interval gives the average, in the same way that dividing total distance by total time gives average speed. Without dividing, you would get a sum, not a typical value.

### Does the average value always occur at some point?

Yes, if f is continuous. The mean value theorem for integrals says that f reaches its average at some point c in the interval. For x squared on [0, 3], the average is 3, reached at x equal to the square root of 3.

## Related concepts

- [Area between curves](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-8)
- [Volume of revolution](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-6)
- [Work as an integral](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-9)
- [Antiderivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-1)

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