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Math Help - [SOLVED] Lower and Upper Riemann sums?

  1. #1
    PTL
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    [SOLVED] Lower and Upper Riemann sums?

    Let f(x) = sin x. Estimate (integral from 0 to pi/2) \int {f(x)} dx by calculating the lower Riemann sums L(f,P_n) and the upper Riemann sums U(f,P_n) for n = 3 with respect to a partition of P_n [0,\frac{\pi}{2}] into n subintervals of equal length. Verify that
    L(f,P_3)<= 1 <= U(f,P_3)
    and explain why L(f,P_n) <= 1 <= U(f,P_n) for all n > 0


    I don't even know what this means!!
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by PTL View Post
    Let f(x) = sin x. Estimate (integral from 0 to pi/2) \int {f(x)} dx by calculating the lower Riemann sums L(f,P_n) and the upper Riemann sums U(f,P_n) for n = 3 with respect to a partition of P_n [0,\frac{\pi}{2}] into n subintervals of equal length. Verify that
    L(f,P_3)<= 1 <= U(f,P_3)
    and explain why L(f,P_n) <= 1 <= U(f,P_n) for all n > 0


    I don't even know what this means!!
    n=3\Rightarrow\Delta{x}=\frac{\pi}{2n}=\frac{\pi}{  6}

    \text{ Let }x_i be the right end point of each subinterval, then

    x_i=(0+\frac{\pi{i}}{6}), therefore

    U=\sum_{i=1}^{3}\sin{\frac{\pi{i}}{6}}*\frac{\pi}{  6}

    Can you proceed?
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by PTL View Post
    I don't even know what this means!!
    I suggest you look for the definitions of an upper/lower Riemann sum in your notes or in your textbook. Once you know the definitions this problem will be easy to solve.
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  4. #4
    Super Member Random Variable's Avatar
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     P_{3} = \{ 0, \frac{\pi}{6} , \frac{\pi}{3}, \frac{\pi}{2} \}

    let  M_{i} be the supremum of f(x) on interval i, and let  m_{i} be the infinum of f(x) on interval i

    then  M_{1} = \frac{1}{2}, \ M_{2} = \frac{\sqrt{3}}{2}, \ M_{3} = 1

    and  m_{1} = 0, \ m_{2} = \frac{1}{2}, \ m_{3} = \frac{\sqrt{3}}{2}

    and since the length of each interval is  \frac{\pi}{6}

     U(f, P_{3}) = \frac {\pi}{6} \Big(\frac{1}{2}+\frac{\sqrt{3}}{2} +1\Big) \approx 1.24

     L(f, P_{3}) = \frac {\pi}{6} \Big(0+\frac{1}{2} +\frac{\sqrt{3}}{2}\Big) \approx 0.715
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    No one in Particular VonNemo19's Avatar
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    Here's a picture of the right sum
    Last edited by VonNemo19; September 19th 2009 at 10:49 PM.
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  6. #6
    Super Member Random Variable's Avatar
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    .
    Last edited by Random Variable; August 2nd 2009 at 03:14 AM.
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  7. #7
    PTL
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    Thanks guys.

    Question - how does one mark a thread as Solved?

    Also - there's a bit at the end of the question, where I'm asked to
    "Explain why L(f,P_n) <= 1 <= U(f,P_n) for all n > 0".
    Can I just say it's because the function is monotonically increasing and therefore the Upper Sum will be an overestimation, and the Lower Sum will be an underestimation?
    Where does the '1' come into that though?
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  8. #8
    MHF Contributor
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    Quote Originally Posted by PTL View Post
    Thanks guys.

    Question - how does one mark a thread as Solved?

    Also - there's a bit at the end of the question, where I'm asked to
    "Explain why L(f,P_n) <= 1 <= U(f,P_n) for all n > 0".
    Can I just say it's because the function is monotonically increasing and therefore the Upper Sum will be an overestimation, and the Lower Sum will be an underestimation?
    Where does the '1' come into that though?
    The 1 is the actual area under the curve, i.e. \int_0^{\pi/2} \sin x \, dx = 1
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  9. #9
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by PTL View Post
    Question - how does one mark a thread as Solved?
    Go to the top of the page, click on "Thread Tools" and then on "Mark this thread as solved".
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  10. #10
    PTL
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    Thanks.
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