Mean Value of a Function

The mean value (or average value) of a function over an interval is a powerful concept in Further Mathematics. It connects integration to the idea of averaging, and provides elegant geometric and physical interpretations.

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1 Definition

The mean value of a continuous function f(x) over the interval [a, b] is:

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

This formula says: divide the total area under the curve by the width of the interval. The result is the height of a rectangle with the same base [a, b] and the same area as the region under the curve.

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2 Geometric Interpretation

Consider the curve y = f(x) between x = a and x = b. The area under this curve is:

A = integral from a to b of f(x) dx

Now imagine a horizontal line y = f_mean such that the rectangle with base (b - a) and height f_mean has exactly the same area A. Then:

(b - a) * f_mean = A

so f_mean = A / (b - a), which matches our formula.