Calculus I: Average Value and Mean Value Theorem for Integrals, Summaries of Calculus

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MATH 12002 - CALCULUS I

§4.4: Average Value & Geometry

Professor Donald L. White

Department of Mathematical Sciences Kent State University

Recall that the average value of a function is found as follows:

Average Value

Let y = f (x) be a continuous function. The average value of f on the interval [a, b] is

fave =

b − a

∫ (^) b

a

f (x) dx.

The average value of a positive function can be described in terms of areas.

Suppose y = f (x) is continuous and f (x) > 0 for a 6 x 6 b, with average value fave = (^) b−^1 a

∫ (^) b a f^ (x)^ dx^ on [a,^ b]. We know that

∫ (^) b a f^ (x)^ dx^ is the area under the graph of^ f^ on [a,^ b].

Since fave = (^) b−^1 a

∫ (^) b a f^ (x)^ dx, this area is also equal to (b^ −^ a)fave.

Observe that (b − a)fave is the area of the rectangle of height fave on [a, b].

f

 -

6

?a (^) b

fave

Hence the average value fave of y = f (x) on [a, b] is the number such that the area under the horizontal line y = fave between x = a and x = b is equal to the area under the graph of y = f (x) between x = a and x = b.

f

 -

6

?a (^) b

fave

f

 -

6

?a (^) b

fave

If you aren’t convinced by those pictures, you can watch the red shaded region move back and forth:

f

 -

6

?a (^) b

fave

If you aren’t convinced by those pictures, you can watch the red shaded region move back and forth:

f

 -

6

?a (^) b

fave

Mean Value Theorem For Integrals

By the Extreme Value Theorem, if y = f (x) is continuous on the closed interval [a, b], then f has an absolute minimum m and an absolute maximum M on [a, b]. Thus m 6 f (x) 6 M on [a, b], and so

m(b − a) 6

∫ (^) b

a

f (x) dx 6 M(b − a),

by a comparison property of definite integrals. Dividing by b − a, we get

m 6

b − a

∫ (^) b

a

f (x) dx 6 M;

that is, m 6 fave 6 M. By the Intermediate Value Theorem, f takes on every y value between m and M, and therefore there is a c with a < c < b such that

f (c) =

b − a

∫ (^) b

a

f (x) dx = fave.

Mean Value Theorem For Integrals

This is the Mean Value Theorem for Integrals:

Theorem

If y = f (x) is a continuous function on [a, b], then there is a number c with a < c < b such that

f (c) =

b − a

∫ (^) b

a

f (x) dx.

The theorem says that a continuous function on a closed interval must take on its average (i.e., mean) value on the interval.