How to apply the mean value theorem for integrals

[misskitty]

Can someone please explain to me how to use this and give an example that we can walk through please? My book doesn't give an example of what it is talking about.

~Kitty

 

[Hurkyl]

Your just talking about the theorem that if $a \leq b$, and f is continuous, then:

$$\int_a^b f(t) \, dt = f(x) (b - a)$$

has a solution for x in the range $a \leq x \leq b$?

An example is easy enough. What's your favorite continuous f? Your favorite a? Your favorite b larger than a? Just plug them in.

Say, we use
f(t) = t²
a = 3
b = 7

Then the mean value theorem says that the equation

$$\int_3^7 t^2 \, dt = x^2 (7 - 3)$$

has a solution with $3 \leq x \leq 7$.