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Average Value of a Function for AP Calculus

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

Mean Value Theorem for Integrals

If f is continuous on [a, b], then there exists a number c in [a, b] such that f(x) dx = f(c)(ba). See Figure 13.1-1.

Mean Value Theorem for Integrals

Example 1

Given verify the hypotheses of the Mean Value Theorem for Integrals for f on [1, 10] and find the value of c as indicated in the theorem.

    The function f is continuous for x ≥ 1, thus:

Example 2

Given f(x) = x2, verify the hypotheses of the Mean Value Theorem for Integrals for f on [0, 6] and find the value of c as indicated in the theorem.

    Since f is a polynomial, it is continuous and differentiable everywhere,

Since only is in the interval [0, 6],

Average Value of a Function on [a,b]

If f is a continuous function on [a, b], then the Average Value of f on [a, b]

.

Example 1

Find the average value of y = sin x between x = 0 and x = π.

Example 2

The graph of a function f is shown in Figure 13.1-2. Find the average value of f on [0, 4].

Average Value of a Function on [a, b]

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

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