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Math Help - general question about integration

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    Member Jskid's Avatar
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    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.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Jskid View Post
    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
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Jskid View Post
    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.
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    Grand Panjandrum
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    Quote Originally Posted by Drexel28 View Post
    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
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