1. If f is Riemann integrable from a to b, and for every rational number r, f(r)=0, then show that the integral from a to b of f(x) is 0. The problem with this question is that you don't know what f is at an irrational. I know that I'm probably supposed to use that rationals are dense in R, but other than that, I'm not sure. 2. Let f(x)= sigma sin nx/(n-1)! where sigma is the sum from n=1 to infinity. Show that the int f(x)dx exists (Riemann integral is from 0 to pi), and evaluate. So, I guess I show that the integral exists because f(x) is pointwise continuous? I'm really confused on this question, and how I can evaluate it. 3. What's a relatively straigt-forward way of proving that if f is riemann integrable, then lim n-> infinity of int f(x) cosnx dx =0, where the integral is evaluated from a to b? Any hints?