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Math Help - What am I doing wrong?

  1. #1
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    What am I doing wrong?

    Integrate f(x)=2x. Lower limit is 1 and upper limit is 2.

    Using the shortcuts, it's easy for me to find F(2) - F(1) = 3.

    However, I was asked to use the limit of Reinmann sums in order to solve the problem...

    \int f(x) dx = \lim_{k\to\infty}\sum_{j=1}^{k}(f(x_j) \Delta x)

    f(x) = 2x

    \Delta x = \frac{2 - 1}{k} = \frac{1}{k}

    x_j = \frac{1}{k} j = \frac{j}{k}

    \sum_{j=1}^{k} 2 \frac{j}{k}  (\frac{1}{k})


    \frac{2}{k^2}\sum_{j=1}^{k}j
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  2. #2
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    Re: What am I doing wrong?

    Quote Originally Posted by Nervous View Post
    Integrate f(x)=2x. Lower limit is 1 and upper limit is 2.
    Using the shortcuts, it's easy for me to find F(2) - F(1) = 3.
    However, I was asked to use the limit of Reinmann sums in order to solve the problem...
    \int f(x) dx = \lim_{k\to\infty}\sum_{j=1}^{k}(f(x_j) \Delta x)
    f(x) = 2x
    \Delta x = \frac{2 - 1}{k} = \frac{1}{k}
    \color{red}x_j = \frac{1}{k} j = \frac{j}{k}

    This is your mistake.

    It should be x_j =1+ \frac{1}{k} j
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  3. #3
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    Re: What am I doing wrong?

    continued...

    \frac{2}{k^2} \frac{k^2 + k}{2} = \frac {2k^2 + 2k}{2k^2} = 1 + \frac {1}{k}

    And obviously...

     \lim_{k\to\infty} 1 + \frac {1}{k} = 1

    So what am I doing wrong?
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  4. #4
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    Re: What am I doing wrong?

    Thanks, just saw your post.

    I had to go to a second post because this tablet wouldn't let me scroll down...
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  5. #5
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    Re: What am I doing wrong?

    How did you get that?
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  6. #6
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    Re: What am I doing wrong?

    Quote Originally Posted by Nervous View Post
    How did you get that?

    To answer your question quite literally: I have been doing Riemann Sums since 1966.

    On [a,b],~\Delta x=\frac{b-a}{n}:
    left-hand sums need: x_j=a+(j-1)\Delta X,~j=1,2,\cdots,n

    right-hand sums need: x_j=a+(j)\Delta X,~j=1,2,\cdots,n

    etc.
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  7. #7
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    Re: What am I doing wrong?

    Quote Originally Posted by Plato View Post
    To answer your question quite literally: I have been doing Riemann Sums since 1966.
    My God! You are almost as old as I am!
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  8. #8
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    Re: What am I doing wrong?

    Quote Originally Posted by HallsofIvy View Post
    My God! You are almost as old as I am!
    I am probably older than you. I am in the class of 1964, Millsaps College. I taught two years of high school in Green Cove Springs, FL.
    Last edited by Plato; December 6th 2012 at 05:27 PM.
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  9. #9
    MHF Contributor MarkFL's Avatar
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    Re: What am I doing wrong?

    Hey, that's in my neck of the woods! Was the school on State Road 16?
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    Re: What am I doing wrong?

    Quote Originally Posted by MarkFL2 View Post
    Hey, that's in my neck of the woods! Was the school on State Road 16?
    Not in my day.
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  11. #11
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    Re: What am I doing wrong?

    Quote Originally Posted by Plato View Post
    Not in my day.
    But we had Friday night dinner at Pappies in Saint Augustine.
    Is it still there? I recall the huge tumblers of iced tea.
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  12. #12
    MHF Contributor MarkFL's Avatar
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    Re: What am I doing wrong?

    I think I know the place you are referring to. There used to be a seafood restaurant near the S.R. 207 and I-95 interchange by that name, but it has since become a tobacco shop I believe. The name is still the same though.
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