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Home Applications of the Integral Averages Theorem 2: Mean Value Theorem for Integrals
 
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Theorem 2: Mean Value Theorem for Integrals

THEOREM 2

(Mean Value Theorem for Integrals)

Let f be continuous on [a, b]. Then there is a point c strictly between a and b where the value of f is equal to its average value,

06_applications_of_the_integral-277.gif

06_applications_of_the_integral-281.gif

Figure 6.5.3

PROOF

Theorem 2 is illustrated in Figure 6.5.3. We can make f continuous on the whole real line by defining

f(x) = f(a) for x < a

and

f(x) = f(b) for x > b.

By the Second Fundamental Theorem of Calculus, f has an antiderivative F. By the Mean Value Theorem there is a point c strictly between a and b at which F'(c) is equal to the average slope of F,

06_applications_of_the_integral-278.gif

But

F'(c) = f(c)

and

F(b) - F(a) = 06_applications_of_the_integral-279.gif,

so

06_applications_of_the_integral-280.gif

Example 2: Average Velocity
Example 3: Average Velocity With Respect to Time and Distance

Home Applications of the Integral Averages Theorem 2: Mean Value Theorem for Integrals

Last Update: 2006-11-05