How do you know what ethical integrity entails? What standard do you use to judge your friends' behavior?
Standards that have been with us for thousands of years, both within religious contexts and without. Do you honestly believe that before Christianity - or before Judaism, for that matter - the ideas set forth formally in the Ten Commandments (among others) were completely alien to humanity? Pre-Judaic cultures were well aware that lying, cheating, stealing, not keeping one's word, etc. ..were WRONG, and they didn't need a formal, monotheistic religion to confirm this for them. And in today's world, atheists are no more or less likely to act unethically than people who are formal members of a religion.
You've nailed the essential problem with humanism. If there is no God, there is no good. We are functionally similar to the other animals. There is no such thing as an evil crocodile. He is what he is, and he does what he does. If there is no God, then there is no reason not to kill, steal, rape, whatever. Allister Crowley said, "the sum of the law is do what thou wilt." Without God, the only two questions are, "do I want to do it?" and "can I get away with it?" I've gotten into this debate way too many times to think I'll persuade anyone, but you've nailed it solidly.