Average Value of a Function Calculator

The average value of a continuous function f over an interval [a, b] is:

f_avg = (1 / (b - a)) * integral from a to b of f(x) dx

Think of it as the height of a rectangle with base (b - a) that has the same area as the region under the curve. That is exactly what the integral divided by the width gives you.

For f(x) = Ax^2 + Bx + C, the exact integral is [Ax^3/3 + Bx^2/2 + Cx] evaluated from a to b, so the average value is just that result divided by (b - a). No approximation needed.

The Mean Value Theorem for Integrals guarantees that for any continuous function, there exists at least one point c in (a, b) where f(c) = f_avg. This calculator also finds that c value for the quadratic case by solving Ax^2 + Bx + C = f_avg using the quadratic formula.

Where this matters: in physics, the average value of a velocity function over a time interval gives average speed. The average value of a force over a displacement gives the equivalent constant force (useful in impulse calculations). In signal processing, the average value of a periodic signal is its DC component.

The average value is not the same as the average of f(a) and f(b) – that would only be exact for linear functions. For curves, the integral correctly weights every point on the domain.