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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:

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 *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 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.

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