Average Value of a Function Calculator

The average value of a function over an interval is the height of the rectangle that has the same area as the region under the curve. You find it by integrating the function over the interval and dividing by its width. This calculator integrates exactly when possible, checks numerically and draws the matching rectangle.

How to use the average value of a function calculator

Type the function.
Enter the start and end of the interval.
Press Calculate to see the average value and the graph.

Formula

f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx

Worked example: x² on [0, 3]

f(x) =: x^2
From x =: 0
To x =: 3

✓ Answer checked

f_{\text{avg}} = 3

Integral: $9$

The green rectangle of height 3 has the same area as the region under the curve.

Step-by-step working (3 steps)

Formula Average value
The average value is the integral divided by the width of the interval.
$f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx$
Integrate
Use an antiderivative.
$\int_{0}^{3} x^{2}\,dx = 9$
Divide by the width
Width = 3.
$\frac{9}{3} = 3$

Formulas used

f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx

How this was checked

✓ Numerical integration gives the same integral.

Frequently asked questions

What is the average value of x² on [0, 3]?

The integral is 9 and the width is 3, so the average is 3.

Is it the same as averaging the endpoints?

No. It averages every value across the interval, not just the two ends.

What is the mean value theorem for integrals?

For a continuous function, there is a point in the interval where f equals its average value.

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