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I Rate of change of area under curve f(x) = f(x)

  1. Jan 2, 2018 #1
    What lead to the equality of the, rate of change of area under curve f(x) = f(x).

    Was it, they were just compared(OR believed to be equal) and mathematically found to be equal. Or when one was integrated or differentiated the other appeared.

    Also I knew, integration was being used since Archimeded and Greeks times.

    Thanks.
     
  2. jcsd
  3. Jan 2, 2018 #2

    PeroK

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    If you take a function at a point ##f(x_0)##. That's the height of the curve at that point. The area between this and another point a short distance to the right is approximately a rectangle of width ##\Delta x##, hence of area ##\Delta A \approx f(x_0) \Delta x##.

    And the rate of change of area is approximately ##\frac{\Delta A}{\Delta x} \approx f(x_0)##

    Using calculus makes this exact, but intuitively the rate of change of area under the curve is simply the height of the curve at that point.
     
  4. Jan 2, 2018 #3
    It look so easy, how is it that only calculus make it possible later , the relation between integration and differentiation.
     
  5. Jan 2, 2018 #4

    PeroK

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    I think it is relatively easy to see. But, calculus is needed to prove it formally and also to put conditions on the function involved - in this case, for example, that the function ##f## is continuous is sufficient. It's not so easy to see intuitively that it works for every continuous function, as some continuous functions are very strange indeed.
     
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