It is not entirely obvious what is meant by the average (or mean) value of a function on an interval. We know how to find the mean of a finite collection of numbers (their sum divided by their number). Needless to say, we run into problems when we want to talk about the mean of all the values of a function on a particular interval, since they are infinite in number.

To find our way out of this conundrum, we recall the definition of the
*n*-th (upper) Riemann sum for the function *f* on the interval
[*a*, *b*]:

U_{n}(f, a, b) = M_{i} |

Note that *U*_{n}(*f*, *a*, *b*) is equal to to the product of *b* - *a* (the length
of the interval) and the mean of the values of *f* at *n* more or less
evenly-spaced points in the interval. Clearly this is a reasonable
approximate mean of the function *f* on the interval [*a*, *b*].

Naturally, the same is true for the *n*th lower Riemann sum. As *n*
gets larger and larger, we might imagine the upper and lower Riemann
sums to approach (one from above, one from below) the product of *b* - *a*
and some "true" mean of the function *f* on [*a*, *b*]. Indeed, this
indicates precisely how we will define the average value, denoted
. We set

= | U_{n}(f, a, b) | ||

= | L_{n}(f, a, b) | ||

= | f (x)dx |

There is a way of seeing graphically that this definition makes sense. An easy
computation shows that the integral of the constant from *a* to *b* is
equal to that of the function *f* (*x*):

dx | = | |_{a}^{b} | |

= | (b - a) | ||

= | f (x)dx |

Thus, is the height of a rectangle of length *b* - *a*
that will have the same area as the region below the graph of *f* (*x*)
from *a* to *b*. In physical terms, if *f* (*t*) represents the velocity
of a moving object, then another object moving with velocity
will travel the same distance between the moments
*t* = *a* and *t* = *b*.

Figure %: Plot of *f* (*x*) and

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