I just need someone to help me get started with this.
Let f:[a,b] -> R (reals) be Riemann Integrable over [a,b] and let alpha be in the real numbers. Define f_alpha : [a + alpha, b + alpha] -> R (reals) by
f_alpha = f(x-alpha). Show that f_alpha is Riemann Integrable and Riemann Integrable of f_alpha(x) dx from a+alpha to b+alpha is equal to the Riemann Integral of f(x) dx from a to b.
Sorry about how it looks...not really sure how to put in all the fancy symbols. There are three ways so far that I can start using, but I'm not really sure on how to actually use them. So, I can use the definition of Riemann Integrable outright, I could use the definition of the Riemann Condition, or I could try to prove it is Darboux Integral and from the Equivalence Theorem it would be Riemann Integrable. But I have no idea how to start. Any help would be awesome.
I think you should just be able to plug f_alpha into the definition of the Riemann integral and verify that it reduces to the integral of f (with the limits of integration modified). Did you try that yet?
I think the first thing we have to do is prove that f_alpha is Riemann Integrable to begin with. To prove this (from what was taught) involves epsilons. I was thinking about assuming that f_alpha is Riemann Integrable and work on the second part of the proof using Liebnitz's Rule, but then again I am not sure.