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Riemann integral is zero for certain sets

  1. Mar 31, 2013 #1
    1. The problem statement, all variables and given/known data
    The question is:

    Let ##\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}##. Prove that if ##E\subset\pi## is a closed Jordan domain, and ##f:E\rightarrow\mathbb{R}## is Riemann integrable, then ##\int_{E}f(x)dV=0##.




    2. Relevant equations

    n/a

    3. The attempt at a solution
    (How to relate the condition it's Riemann integrable to the value is ##0##? The textbook I use define ##f## is integrable on ##E## iff ##\;\;\;\;(L)\int_{E}fdV=(U)\int_{E}fdV##)
     
  2. jcsd
  3. Mar 31, 2013 #2

    Fredrik

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    What is the definition of "closed Jordan domain"?

    Regardless of what the answer to that is, the strategy here should definitely be to prove that given ##\varepsilon>0##, there's an upper sum U and a lower sum L such that ##-\varepsilon<L<U<\varepsilon##. You may want to try this for an especially simple choice of E and f before you try the general case.
     
    Last edited: Mar 31, 2013
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