Average Value of a Function Calculator
Find the average value of a function over an interval — enter f(x) and the bounds a and b to get (1/(b − a)) × the definite integral, computed numerically.
Computes the average value of f on [a, b] = (1/(b − a)) ∫ab f(x) dx, with the integral found numerically (Simpson’s rule). Allowed: + − * / ^, parentheses, x, sin, cos, tan, ln, log, sqrt, exp, abs, pi, e.
How to find the average value of a function
The average value of a continuous function over an interval is the definite integral divided by the length of the interval. Geometrically it’s the height of a rectangle on [a, b] with the same area as the region under the curve.
average value = (1 ÷ (b − a)) × ∫ from a to b of f(x) dx
For example, the average value of f(x) = x² on [0, 3] is (1 ÷ 3) × ∫₀³ x² dx = (1 ÷ 3) × 9 = 3. By the Mean Value Theorem for integrals, a continuous function reaches its average value at least once on the interval — here, x² equals 3 at x = √3 ≈ 1.73.
Working with plain numbers instead of a function? Use the average calculator for the mean, median and mode.
Frequently asked questions
How do you find the average value of a function?
Integrate the function over the interval and divide by the interval length: average = (1/(b − a)) × ∫ from a to b of f(x) dx. For f(x) = x² on [0, 3], the integral is 9 and 9 ÷ 3 = 3, so the average value is 3.
What is the average value of a function?
It is the constant height of a rectangle over [a, b] that has the same area as the region under the curve. By the Mean Value Theorem for integrals, a continuous function actually attains this average value at least once on the interval.
What can I enter as f(x)?
Use x, numbers and + − * / ^, parentheses, and functions like sin, cos, tan, sqrt, ln, log, exp and abs, plus the constants pi and e. The integral is computed numerically (Simpson’s rule), so a function with a break between a and b may give an inaccurate result.