Is differentiation more fundamental than integration? I ask because every definition I heard for integration, has to do with the reverse process of differentiation.
That's because you've never heard the real definition. I don't mean this in a mocking way, it's just the truth. The Fundamental Theorem of Calculus, which is surely what you refer to, states (roughly) that if then . But, what if doesn't satisfy the hypotheses? The definition of integrability etc. is kind of 'high-tech' and 'notation laden'. See Apostol or Rudin for a full explanation.