How Does the Integral Mean Value Theorem Link to Fundamental Calculus Concepts?

[daudaudaudau]

http://en.wikipedia.org/wiki/Mean_value_theorem#First_mean_value_theorem_for_integration"

Take a look at the Wikipedia proof. Now, wouldn't it be easier to prove it like this:
The ordinary mean value theorem says that
$$G(b)-G(a)=(b-a)G'(\xi)$$

And the fundamental theorem of calculus says that
$$G(b)-G(a)=\int_a^b G'(x)dx$$

So the conclusion is
$$\int_a^b G'(x)dx=(b-a)G'(\xi)$$

 

[Office_Shredder]

You proved the case where $$\phi = 1$$ identically. Notice the wikipedia proof covers a far broader case than you did.

Also, you probably used the mean value theorem to prove the fundamental theorem of calculus, so this is circular

 

[daudaudaudau]

I see.