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Homework Help: Need some help with Riemann Sums.

  1. Dec 1, 2008 #1
    Need some urgent help with Riemann Sums.

    1. The problem statement, all variables and given/known data

    PART A:

    In all of this question, let I = [tex]\int ^{2}_{-2} f(x)dx [/tex] [tex]where f(x) = -2x + 1 [/tex]

    Evaluate I.


    PART B:

    Use the defintion of the definite integral to evaluate I.
    i.e Riemann Sum.


    2. Relevant equations



    3. The attempt at a solution

    PART A:

    I = [-(2)^2 + (2)] - [-(-2)^2 + (-2)]
    I = -2 + 6 = 4


    Part B is what i am having trouble with.

    Xk = a + k([tex]\Delta x[/tex])
    a = -2

    and

    [tex]\Delta x[/tex] = b-a/n = 2- (-2)/n = 4/n

    so then

    Xk = -2 + 4k/n

    Now the Reimann sum is:

    [tex]\sum f(Xk)(\Delta x)[/tex] = [tex]\sum [-2(-2 + 4k/n) + 1](4/n)[/tex]
    = 4/n [tex]\sum 4 - 8k/n + 1 [/tex]
    = 4/n [tex]\sum -8k/n + 5[/tex]
    = 4/n x -8/n [tex]\sum k[/tex] + [tex]\sum 5[/tex]
    = -32/n2 x [(n)(n+1)/2] + 5n

    Now when i take the limit approaching infinity for that, it doesnt give me 4 as an answer unless im doing something wrong.
    This is where i am stuck at.
     
  2. jcsd
  3. Dec 1, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Need some urgent help with Riemann Sums.

    Your last two lines start to look pretty goofy. The limit of (4/n)*(5n-8*n(n+1)/2) is 4. Then parentheses start disappearing and numbers are magically factored out. Check it again.
     
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