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A little confused about integrals

  1. Jan 8, 2015 #1
    I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function?

    If an integral is the area under a curve, why do we even have to find the anti derivative at all?
     
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  3. Jan 8, 2015 #2

    PeroK

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    An integral is the area under the curve of the (original) function.

    This area is, however, closely related to the anti-derivative of the function. This is the Fundamental Theorem of Calculus.
     
  4. Jan 8, 2015 #3

    Ray Vickson

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    The antiderivative (if available) is useful because if ##A(a,z)## is the area under the curve ##y = f(x)## from ##x=a## to ##x=z## (with ##a## some fixed number), and if ##F(x)## is the antiderivative of ##f(x)##, then we have:
    [tex] \text{area} = A(a,z) = F(z) - F(a)[/tex]
    (Fundamental Theorem of Calculus).

    So, if you know the antiderivative function you can use it to calculate the area. That is a lot easier than splitting up the interval ##[a,z]## into 100 billion little intervals and adding up the 100 billion little rectangular areas (which would give a good approximation to the actual area---not necessarily the exact area).
     
  5. Jan 19, 2015 #4

    Svein

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    Integration can be used to find the area under a curve, but that is just one trivial function of the integration tool.

    As to anti-derivatives: The definition is that F is the anti-derivative of f if dF/dx = f. This has one important consequence:

    F is differentiable. f does not have to be. I fact, f does not even have to be continuous.
     
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