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Average Value of a Function

Also known as mean value of a function · f average · average height of a curve · mean value theorem for integrals · average value over an interval

favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b - a}\int_{a}^{b} f(x)\,dx

Units aren’t used in this calculation: every value is a plain number.

Worked example: x² on [0, 3]: integral 9 → average 3. Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.

See also
Definite Integral of a Power Function: Where the integral usually comes from, when f is a power of x.Slope Between Two Points: The OTHER average: (f(b) − f(a))/(b − a) is the average rate of change, a slope, not an average height.Arithmetic Mean of Two Numbers: The same idea for plain numbers: add them up and divide by how many. Here the integral is the total and the width b − a is the count.
Learning zone

Average Value of a Function explained

The average value of a function over an interval is the height of the rectangle that has the same width and the same area as the region under the curve: the definite integral divided by the width b − a. It is the continuous version of adding up a list and dividing by how many there are. Worked example: sin x from 0 to π. The integral is 2, the width is π, so the average height is 2/π = 0.6366. A rectangle 0.6366 tall and π wide has area 2, the same as the arch of the sine.

The classic mistake is confusing the average value with the average rate of change. Over the same interval, (sin π − sin 0)/π = 0: the curve ends where it started, so its average slope is zero, while its average height is 0.6366. A related slip is averaging the two endpoint values, (f(a) + f(b))/2, which for the sine is also 0. The mean value theorem for integrals adds that a continuous function actually takes its average value somewhere on the interval. In practice this is how a varying quantity is boiled down to one number: the average temperature over a day, average velocity as the average value of the velocity, and the RMS voltage of an AC supply, whose square comes from the average value of sin², which is ½ over a full cycle.

Average Value of a Function formula

favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b - a}\int_{a}^{b} f(x)\,dx
Where
  • favgf_{\text{avg}}= Average value of f on [a, b]
  • ∫abf(x) dx\int_{a}^{b} f(x)\,dx= Definite integral of f from a to b
  • aa= Start of the interval a
  • bb= End of the interval b

Missing one of these? Work it out first, then come back

Revised 16 October 2026
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formula.expert. (2026, October 16). Average Value of a Function Formula & Calculator. https://formula.expert/solvers/average-value-of-function

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