# Thread: Riemann-Stieltjes Integral

1. ## Riemann-Stieltjes Integral

Hello!

I am working on this problem:

Suppose $\displaystyle f$ and $\displaystyle g$ are continuous and positive on some interval $\displaystyle [a,b]$.

I am trying to show that $\displaystyle \exists\ \zeta \in [a,b]$ such that $\displaystyle \int_a^bf(x)g(x)dx = f(\zeta)\int_a^bg(x)dx$

All I have so far is that since both $\displaystyle f$ and $\displaystyle g$ are continuous, then their product is continuous (and therefore Riemann-integrable) but I'm not sure how to select this zeta. Whole thing looks like some sort of mean value theorem for products of functions?!

Any help appreciated. Thanks!!

2. Originally Posted by matt.qmar
Hello!

I am working on this problem:

Suppose $\displaystyle f$ and $\displaystyle g$ are continuous and positive on some interval $\displaystyle [a,b]$.

I am trying to show that $\displaystyle \exists\ \zeta \in [a,b]$ such that $\displaystyle \int_a^bf(x)g(x)dx = f(\zeta)\int_a^bg(x)dx$

All I have so far is that since both $\displaystyle f$ and $\displaystyle g$ are continuous, then their product is continuous (and therefore Riemann-integrable) but I'm not sure how to select this zeta. Whole thing looks like some sort of mean value theorem for products of functions?!

Any help appreciated. Thanks!!
You are correct this is a mean value theorem for integrals

See this link

Mean value theorem - Wikipedia, the free encyclopedia

The proof on the wiki page shows you how to select the value you wish.