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Math Help - Correlation between integrals and differentials

  1. #1
    Senior Member Paze's Avatar
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    Correlation between integrals and differentials

    Hi MHF!

    I'm wondering...Can you tell me why a definite integral is the opposite of a derivative? How can an area be the opposite of a slope? Are we going from first to 2nd dimension somehow...? If so, why?

    Thanks!
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  2. #2
    MHF Contributor
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    Re: Correlation between integrals and differentials

    The slope of the area S(x) under the graph of f(x) is again f(x). Indeed, the slope of the area is the speed with which the area increases: (S(x+h) - S(x))/h. But S(x+h) ≈ S(x) + f(x) * h, so (S(x+h) - S(x))/h ≈ f(x).
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  3. #3
    MHF Contributor
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    Re: Correlation between integrals and differentials

    Are you familiar with the Fundamental Theorem of Calculus? If not, I would recommend checking out Wikipedia. It offers a good visualization of the theorem.
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