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The Fundamental Theorem of Calculus

  1. Jul 26, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm having trouble wrapping my head around this concept. I understand integration and differentiation individually. I even understand the algebraic manipulations that reveals their close relationship. However, the typical geometric interpretation of a 1-D curve being the derivative of the area function below it seems odd to me. I'm trying to get an intuitive understanding of how a 1-D curve, say A'(x) = f(x) , is the derivative of an area function, A(x). I think I might know where my source of confusion lays: When dealing with derivatives, I'm used to visualizing in terms of tangent lines drawn to a 1-D curve. It seems weird to apply it to an area function, which is more irregular polygon than it is 1-D curve. I hope my concerns make sense.

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Jul 26, 2015 #2

    BvU

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    Googling doesn't help ?
     
  4. Jul 26, 2015 #3
    try googling

    mit Session 47: Introduction of the Fundamental Theorem of Calculus

    and watch that
     
  5. Jul 26, 2015 #4
    + S[x+dx] = S[x] + f(x)dx. Obvious from geometry. => f(x)=S'(x)
     
  6. Jul 26, 2015 #5
    Great video, but it didn't quite answer my question. I might not have been able to articulate my question that well over the internet. Oh, well. Thanks, anyhow.
     
  7. Jul 27, 2015 #6
    I think the missing link between integration and differentiation is primitiv(-isation,-ization, -ation, ..., I don't know). It is the 'reciprocal' operation to differentiation: ##\phi## is a primitive of a piecewise continuous fonction ##f## if and only if ##\phi## is differentiable on the domain of ##f## and ##\phi' = f##.
    The set of all primitives of ##f## vary by an additive constant. So all primitives of ##f## have the form ##\phi = \phi_0 + C##, where ##\phi_0## is a particular primitive of ##f##.
    It happens that the function ##\phi_0(x) :=\int_{x_0}^x f(t) dt ## is a primitive of ##f## wherever ##f## is continuous (just show that if ## f## is continuous in ##a##, ## \frac{\phi_0(x) - \phi_0(a)}{x-a} \rightarrow f(a) ## as ## x\rightarrow a## ). I think it explains a little bit the link between integration and differentiation.


    The link between the area under the curve of ##f##, defined on ##[a,b]##, and its integral, comes from the construction of the integral of piecewise continuous functions from the integral of step-functions on ##[a,b]##, which are homogenous to an area
     
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