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Fundamental theorem question

  1. Jan 26, 2014 #1

    joshmccraney

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    hey pf!

    i'm trying to get a geometric understanding of the fundamental theorem: [tex]\int_a{}^{b}f'(x)dx=f(b)-f(a)[/tex] basically, isn't the above just saying that if we add up a lot of slopes on a line at every point we will get the difference of the y values?

    thanks! feel free to add more or correct me
     
  2. jcsd
  3. Jan 26, 2014 #2

    mfb

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    That's one way to express the theorem in words, indeed.
     
  4. Jan 26, 2014 #3

    PeroK

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    To help understand the fundamental theorem, first, re-arrange it a little:

    [tex]f(b) = f(a) + \int_a{}^{b}f'(x)dx[/tex]
    Now, imagine going from a to b in a number of small steps: a to x1 to x2 ... to b:

    [tex]f(x_1) \approx f(a) + f'(a)dx_1 \ (dx_1 = x_1 - a)[/tex]
    [tex]f(x_2) \approx f(x_1) + f'(x_1)dx_2 = f(a) + f'(a)dx_1 + f'(x_1)dx_2 \ (dx_2 = x_2 - x_1)[/tex]

    So that:
    [tex]f(b) \approx f(a) + \sum f'(x_{i-1})dx_i[/tex]

    And, the Integral is the limit of this as the steps get smaller:
    [tex]f(b) = f(a) + \int_a{}^{b}f'(x)dx[/tex]
     
  5. Jan 26, 2014 #4

    joshmccraney

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  6. Jan 27, 2014 #5

    FactChecker

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    Not just slope. Slope times dx, which is dy. So it is adding up many dys to get from y(a) to y(b).
     
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