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Fundamental theorem of calculus

  1. Aug 28, 2012 #1
    [itex]\frac{d}{dx} \int_a^b f(x) dx=f(b)[/itex]

    This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.
     
  2. jcsd
  3. Aug 28, 2012 #2

    Mark44

    Staff: Mentor

    What you have is incorrect, assuming that both a and b are constants.
    [tex]\frac{d}{dx} \int_a^b f(x) dx =0[/tex]

    The way this is usually presented is like so:

    [tex]\frac{d}{dx} \int_a^x f(t) dt =f(x)[/tex]
     
  4. Aug 28, 2012 #3

    Mark44

    Staff: Mentor

    For an explanation, let's assume that F(x) is an antiderivative of f(x). IOW, F'(x) = f(x).
    Then
    $$ \int_a^x f(t) dt = F(x) - F(a)$$
    So $$ d/dx \int_a^x f(t) dt = d/dx( F(x) - F(a)) = F'(x) - 0 = f(x)$$
     
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