Calculus Of One Real Variable – By Pheng Kim Ving Chapter 12: Applications Of The Integral – Section 12.1: The Mean-Value Theorem For Integrals 12.1 The Mean-Value Theorem For Integrals

 1. Average Values Of Functions  # Fig. 1.1

Average value of a function is limit of average of n function
values as n approaches infinity.

## Definition 1.1  Indeed it does. # Fig. 1.2

Average value of f(x) = y = 2x on [1, 5] is 6.

Example 1.1

Find the average value of f(x) = x2 on [1, 4]. Show that this average value is the height of a rectangle with base width
equal to 4 – 1 = 3 and area equal to the definite integral of f over [1, 4].

#### Solution # Refer to Fig. 1.3. We have: Consider the rectangle with its lower base being the interval [1, 4] and its height equal to 7. Its base width is 4 – 1 = 3, so
its area is (3)(7), which is equal to the definite integral of f over [1, 4]. Thus it's the required rectangle.

EOS 2. The Mean-Value Theorem For Integrals

The average value of S1 = {1, 2, 3} is (1 + 2 + 3)/3 = 2, which belongs to S1, while that of S2 = {1, 2, 9} is (1 + 2 + 9)/3
= 4, which doesn't belong to S2. The average value of a collection of a finite number of quantities may or may not belong
to that collection. Now for a function f that's continuous on [a, b], it of course has an infinite number of values on [a, b].
We'll now see that the average value of f, which is the average value of the collection of all the values of f on [a, b], is
also a value of f, ie, belongs to that collection, provided f is continuous on [a, b]. This means that the average or mean
value of f continuous on [a, b] is attained at some point in [a, b]. This property constitutes the mean-value theorem for
integrals. Recall that the mean-value theorem for derivatives is the property that the average or mean rate of change of a
function continuous on [a, b] and differentiable on (a, b) is attained at some point in (a, b); see Section 5.1 Remarks 5.1.

Theorem 2.1 – The Mean-Value Theorem For Integrals

 If f is continuous on [a, b] then there exists c in [a, b] such that: See Fig. 2.1.

Proof A similar argument leads to the same conclusion if m > M.
EOP Fig. 2.1 ### Remarks 2.1

i. The theorem says that the average value of f on [a, b] is attained on [a, b], ie, it's one of the values of f on [a, b]. ### Example 2.1

Let f(x) = 5x4. + 2. Determine c such that f(c) is the average value of f on [–1, 2].

#### Solution

The average value of f on [–1, 2] is: EOS

 3. An Alternative Proof

Define the function F by: for all x in [a, b]. By the fundamental theorem of calculus (see Section 9.4 Theorem 2.1), f is the derivative (or rate of
change) of F. So the average value of f on [a, b] is the average rate of change of F on [a, b], and the value of f at a
point in [a, b] is the instantaneous rate of change of F at that point. Using this observation we can prove the mean-value
theorem for integrals by applying the mean-value theorem for derivatives to F. This alternative proof is given in Problem
& Solution 5
.

##### Problems & Solutions

1. Find the average value of f(x) = 2x on [0, 3].

Solution  2. Calculate the average value of g(x) = 1 – x2 on [–2, 2].

Solution   Solution  4. Let f(x) = 3x2. Determine c such that f(c) is the average value of f on [0, 2].

Solution

The average value of f on [0, 2] is:   Solution

By the fundamental theorem of calculus, F is differentiable on [a, b]. So, applying the mean-value theorem for
derivatives to F on [a, b], there's c in [a, b] such that: 