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:
f_{avg} = f (x)dx
To see why this is the case, consider this form of the equation:
f_{avg}(b - a) = f (x)dx
The left side of the equation is the area of rectangle with base of (b - a) and height of
f_{avg}. 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:
The equation for the average value is a statement of the intuitive fact that if we construct
a rectangle with the height f_{avg} 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.