I was explaining to a friend about some differences between the Riemann integral and Lebesgue integral. He asked what an integral is and the best I could come up with was "it defines a notion of area" but I don't think that's good enough. There's Riemann integral, Lebesgue, Darboux, Haar, line, surface, double, multiple, etc. So in its most general case, what is an integral? I haven't been able to find widely accepted axiomatic treatment of integration. Could I just say that the integral of all functions is 0 and call it the "johnqwertyful integral"? Or call any object or function I can think of an integral?