1. ## general question about integration

Is differentiation more fundamental than integration? I ask because every definition I heard for integration, has to do with the reverse process of differentiation.

2. Originally Posted by Jskid
Is differentiation more fundamental than integration? I ask because every definition I heard for integration, has to do with the reverse process of differentiation.
No, what you describe is a pedagogical device. They are more or less equally fundamental.

CB

3. Originally Posted by Jskid
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 $f=F'$ then $\displaystyle \int_a^b f(x)\text{ }dx=F(b)-F(a)$. But, what if $f$ 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.

4. Originally Posted by Drexel28
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 $f=F'$ then $\displaystyle \int_a^b f(x)\text{ }dx=F(b)-F(a)$. But, what if $f$ 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.
Also the ideas behind the definite integral pre-date those behind the derivative by something like 1800 years.

CB