The geometric interpretation

The average value has a very telling geometric reading, which helps you never forget the formula. It answers the question: what height of rectangle would have the same area as the region under the curve?

Take μ = 1/(b−a) · ∫ f again. Multiplying both sides by (b − a):

   μ · (b − a)  =  ∫ (from a to b) f(x) dx
   (height × width)      (area under the curve)

In other words, μ is the height of the rectangle with base [a, b] whose area equals exactly the area under the curve.

   f(x)
    |      __--‾‾--__
    |    _/          \_        <- area under the curve
    |___/______________\___
    | . . . . . . . . . . .    <- rectangle of height μ,
    |μ|                   |        SAME area
    +--+---------------+--+--> x
       a               b

We "flatten" the curve into a rectangle with the same area: its height is the average value. This picture explains why μ always lies between the minimum and the maximum of f on the interval: the rectangle can be neither lower than the lowest point of the curve, nor higher than the highest.

   min(f)  ≤  μ  ≤  max(f)        (always)

This interpretation turns an abstract formula into a concrete picture: the average value is the height that equalizes the area. It prepares the mean value theorem, which states that this height is indeed reached by the function.