Average Value of a Function Calculator
Find the average value of f(x) = Ax² + Bx + C over any interval using the mean value theorem for integrals.
Also locates the c where f(c) equals that average.
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) = Ax² + Bx + C, the exact integral is [Ax³/3 + Bx²/2 + Cx] evaluated from a to b, so the average value is that result divided by (b − a). No approximation is involved, which is why the answer comes out to full precision rather than as a numerical estimate.
The Mean Value Theorem for Integrals guarantees that for any continuous function there is at least one point c in (a, b) where f(c) = f_avg. This calculator finds that c for the quadratic case by solving Ax² + Bx + C = f_avg with the quadratic formula. A parabola can cross its own average height twice inside the interval, so you sometimes get two values of c, and both are correct.
Worked example: f(x) = x² over [0, 4]
Integral = [x³/3] from 0 to 4 = 64/3 ≈ 21.3333 Width = 4 − 0 = 4 f_avg = 21.3333 ÷ 4 = 16/3 ≈ 5.3333
Now find c: x² = 16/3, so c = √(16/3) ≈ 2.3094, which sits inside [0, 4] as the theorem promises.
Notice what the shortcut would have given you. f(0) = 0 and f(4) = 16, so averaging the two endpoints gives 8. The true average is 5.3333. That is a 50% overestimate, and it happens because the parabola spends most of the interval well below its final value. Averaging the ends only works for straight lines, where the curve rises at a constant rate. For anything that bends, the integral is doing real work by weighting every point on the domain, not just the two you can see.
Where this matters: in physics, the average value of a velocity function over a time interval gives average speed. The average of a force over a displacement gives the equivalent constant force, which is what impulse calculations need. In signal processing, the average value of a periodic signal is its DC (Direct Current) component.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.