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*)(*b* – *a*). See Figure 13.1-1.

### 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*) = *x*^{2}, 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].

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

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