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Summation of rectangular areas (calculus) problem.

  1. Oct 9, 2004 #1
    Good evening. I'm having a little difficulty with the summation of rectangular areas when finding the area under a curve.

    Question:
    Using summation of rectangles, find the area enclosed between the curve y = x^2 + 2x and the x-axis from x=0 to x=3.

    Well, I start by dividing the interval (from x=0 to x=3) by n equal parts to find the width of each rectangular area.
    =3/n

    Then I begin using sigma notation (I'm new at this)
    Sum of rectangular areas
    [tex]= \sum_{k=1}^n\ f(x) * (3/n)[/tex]
    [tex]= \sum_{k=1}^n\ [(k * 3/n)^2 + 2(k * 3/n)] * (3/n)[/tex]
    [tex]= \sum_{k=1}^n\ [k^2 * (3/n)^2 + 2 * k * (3/n)] * (3/n)[/tex]
    [tex]= \sum_{k=1}^n\ [k^2 * (3/n) + 2k] * (3/n)^2[/tex]
    [tex]= 9/n^2\sum_{k=1}^n\ [k^2 * (3/n) + 2k)][/tex]

    Now my main problem is that I'm trying to isolate the k^2 so that I can write out the summation formula for it and then go to limits and discover the area under the curve.

    eg
    [tex]\sum_{k=1}^n\ k^2[/tex]
    would become
    n(n+1)(2n + 1) / 6
    and I could go straight to the limits
     
    Last edited: Oct 9, 2004
  2. jcsd
  3. Oct 9, 2004 #2

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    I'm not sure what your difficulty is - it appears you're right on the mark!
     
  4. Oct 10, 2004 #3
    The difficulty is that I don't know -how- to isolate the k^2 :(

    I know very little about using this sigma notation, the only rule I know is that when you have sigma(ak + b) you can rewrite as:
    sigma(ak) + sigma(b)
    =(a)sigma(k) + (b)sigma(1)

    where a and b are constants.

    for me, the problem is that there is a variable on the other side of the addition--2k that is...

    Again, I know very little of the rules of sigma notation. Perhaps there is a way to rewrite this? I'm not wanting the answer for the area--rather just a more manageable way for me to put it to limits ;)

    Or is it already to go??? I had just figured I could "do more" to it.

    Thanks again!
     
  5. Oct 10, 2004 #4
    Could you experts tell me if what I'm doing below is "legal" in terms of math

    [tex]9/n^2\sum_{k=1}^n\ k^2 * (3/n) + 2k[/tex]
    becomes
    [tex]= 9/n^2[\sum_{k=1}^n\ k^2 * (3/n) + \sum_{k=1}^n\ 2k][/tex]
    [tex]= 9/n^2 [(3/n)\sum_{k=1}^n\ k^2 + 2\sum_{k=1}^n\ k][/tex]

    And then the first sigma notation would become
    n(n + 1)(2n + 1)
    --------------
    6

    and the second sigma notation would become
    n(n + 1)
    --------
    2

    and I could then expand the 9/n^2 on it and evaluate the limits? This is probably really stupid question but yesterday was the first day I worked with this notation :D
     
  6. Oct 14, 2004 #5

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    Single,

    Sorry for the late reply - I lost track of you there!

    What you did is fine and you're basically done. Now just ask yourself what the limiting value of your expression is as you let n go to infinity.
     
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