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Help with Proof on Integration

  1. Nov 15, 2012 #1
    I was assigned this problem in class. My instructor said it was a very popular theorem, but I cannot find it in my book or online. I am clueless on what to do. I would appreciate the help.

    Let f(x) be bounded and integrable on [a, b]. Assume that g(x) differs from f(x) on only finitely many points in the domain. Show that g(x) is integrable. Moreover, show that ∫f(x)dx = ∫g(x)dx (Both integrals are from b to a).
  2. jcsd
  3. Nov 15, 2012 #2


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    welcome to pf!

    hi tomhawk24! welcome to pf! :smile:

    start with the definition

    which definition of integral (or integrable) are you using?
  4. Nov 15, 2012 #3
    Well we are working mainly on the Fundamental Theorem of Calculus right now.
  5. Nov 15, 2012 #4


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    ok, we'll start with that, then …

    what does the fundamental theorem of calculus say? :smile:
  6. Nov 16, 2012 #5
    Starting from the area interpretation of the integral, answer this question: If I take finitely many points out of the graph of a curve f(x) and place them at some other y-coordinate, would the function still be integrable? What would be its integral?

    Tip: Does a point have dimensions, or does a line have width? What is the area of a rectangle?
  7. Nov 16, 2012 #6


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    First show
    ∫(f(x)-g(x))dx =0
    then use linearity
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