Average value of a function on an interval
Total accumulated output divided by interval length—the continuous counterpart of an arithmetic mean.
LaTeX article Updated July 11, 2026
Why divide by interval length
A longer interval naturally accumulates more area even when the function’s typical height is unchanged. Dividing by b − a removes that duration or width effect.
Units confirm the formula: output-units times input-units, divided by input-units, returns output-units.
Geometric interpretation
The average value is the height of a rectangle with the same signed area as the region under the function over the interval.
If the function is continuous, the Integral Mean Value Theorem guarantees at least one point where the function actually equals its average.
Signed versus physical averages
For velocity, the average value gives average velocity, not average speed. If direction changes and speed is requested, average the absolute value of velocity instead.
Worked example
Common mistakes
- Forgetting the factor 1/(b−a).
- Using endpoint average instead of function average.
- Confusing average velocity with average speed.
Keep these ideas
- Average equals accumulation divided by interval length.
- The result has the same units as the function.
- Interpret signs according to the modeled quantity.