The Average Value of a Function

One of the uses of the definite integral is that it can help us find the average value of a function on an interval [a, b]. The formula for the average value on an interval [a, b] is as follows:

favg = f (x)dx    

To see why this is the case, consider this form of the equation:

favg(b - a) = f (x)dx    

The left side of the equation is the area of rectangle with base of (b - a) and height of favg. The right side of the equation is the area under the curve of f over the interval with length (b - a). These areas are depicted below:

Figure %: Two separate, equal areas

The equation for the average value is a statement of the intuitive fact that if we construct a rectangle with the height favg and width (b - a), its area should be the same as the area under curve from a to b.

The Mean Value Theorem for Integrals

The mean value theorem for integrals states the following: if f is a continuous function on [a, b], there exists at least one c on [a, b] such that

f (c) = f (x)dx    

In other words, the MVT for integrals states that every continuous function attains its average value at least once on an interval.