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Limit Sum Integer Method

  1. Jul 30, 2005 #1


    How is this problem solved using the Limit Sum Integer method?

    [tex]\int_{2}^{10} x^6 \; dx[/tex]

     
  2. jcsd
  3. Jul 30, 2005 #2

    lurflurf

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    Homework Helper

    Limit Sum Integer method is an odd name.
    I assume you mean as a Reiman sum. Usually the Reiman sum is calculated with even spacing.
    [tex]\int_{2}^{10} x^6 \; dx=\lim_{n\rightarrow\infty}\sum_{i=1}^n (x^*_i)^6{\Delta}x_i=\lim_{n\rightarrow\infty}\sum_{i=1}^n (2+i(10-2))^6\frac{(10-2)}{n}[/tex]
    Thus all that is needed to work through to the end is the ability to do sums of polynomials.
     
    Last edited: Jul 30, 2005
  4. Jul 30, 2005 #3

    HallsofIvy

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    Of course, to anyone who actually knows how to do an integral,
    [tex]\int_2^{10}x^6dx= \frac{1}{7}x^7[/tex] evaluated between 2 and 10. You can use that to check your work.
     
    Last edited: Jul 31, 2005
  5. Jul 31, 2005 #4
    Riemann Sum...


    Should not this Riemann sum actually be:
    [tex]\lim_{n\rightarrow\infty}\sum_{i=1}^n \left( 2 + \frac{i(10-2)}{n} \right)^6 \frac{(10-2)}{n}[/tex]

    This was my approach:
    [tex]\int_2^{10} x^6 dx = \lim_{n \rightarrow \infty} \frac{8}{n} \sum_{i = 1}^n x^6 = \lim_{n \rightarrow \infty} \sum_{i = 1}^n \left( 2 + \frac{8i}{n} \right)^6 \cdot \frac{8}{n}[/tex]

    This Riemann sum must be expanded before one can sum the polynomial?
     
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