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Introduction to Integrals

Average Value and Second Fundamental Theorem

Problems for "The Definite Integral"

Average Value and Second Fundamental Theorem, page 2

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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:

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

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.