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Math Help - Riemann Integration

  1. #1
    xyz
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    Riemann Integration



    This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
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  2. #2
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    I believe this what they call a improper integral, and because it is obviously undefined at 0, for something like this I believe you will have to do

    \lim_{x\to0+}{\int_B}^1\frac{1}{x}

    This should get you started
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  3. #3
    xyz
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    What is B in the integral you entered? I don't understand that.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by xyz View Post


    This is an integrable function. However, how is this integral solved? I can perfectly visualize this function. I don't see how the value of integral can be anything besides 0. But I cannot put in words or show work as to how the integral is solved. Please help
    Note that over any subinterval [a,b]\subset [0,1] that \inf_{x\in[a,b]}f(x)=0....so
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  5. #5
    MHF Contributor chisigma's Avatar
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    Is f(x)=0 everywhere with the exception of x=\frac{1}{n} , n \in \mathbb{N} so that you can devide the integration interval in subintervals of the type...

    \frac{3}{4}<x\le 1 , \frac{5}{12} < x \le \frac{3}{4}, \dots , \frac{n+\frac{3}{2}}{n+2}< x \le \frac{n+\frac{1}{2}}{n+1} , \dots

    ... so that in each interval is  f(x)=0 with the only exception of x=\frac{1}{n}, i.e. in only one point. In each interval the function is Riemann integrable and is...

    \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0 (1)

    From (1)...

    \int_{0}^{1} f(x)\cdot dx = \sum_{n} \int_{x_{n+1}}^{x_{n}} f(x)\cdot dx=0 (2)

    Kind regards

    \chi \sigma
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  6. #6
    xyz
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    Thanks a lot for all the help. I already got the idea from Drexel's hint. It was easy after that. But your solution is also very interesting!
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