I had a question about Taylor's theorem proof in here:

and the key point in the answer given to me was this:

there's $c\in(a,b)$ such that

$$\int_a^b f(x)g(x)dx=f(c)\int_a^b g(x)dx$$

Can anyone help me identify this rule? Thank you! =)


Mean value theorem for integration. In case you want to know how this name came about: taking $g=1$, we see that there is $c$ with $$f(c)=\frac{1}{b-a}\int_a^b f \ dx,$$ the right hand side of which is called the integral mean of $f$ over $[a,b]$. (These integrals are fairly important, e.g. because it appears in the Lebesgue differentation theorem; in fact so important that there is even a standard notation. Well it should be without the absolute value bars, but somehow no code seems to work here.)

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  • $\begingroup$ Thank you for the additional info, appreciate it! =) I am very interested =) $\endgroup$ – jjepsuomi Jul 15 '14 at 19:14
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    $\begingroup$ @jjepsuomi: no problem. I thought I should add some information apart from the name itself. $\endgroup$ – user164074 Jul 15 '14 at 19:16
  • $\begingroup$ @user164074 and a good idea it was ;D thank you! =) $\endgroup$ – jjepsuomi Jul 15 '14 at 19:17

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