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Homework Help: Integral involving population density

  1. Aug 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Hey all I've got a problem to do with population density. It asks for you to show the number of people living between arbitrary points a and b from the centre of the town is equal to:

    [tex]P(a,b)=2\pi\int^{b}_{a}rf(r)dr[/tex]

    Where f(r) is the population density. Note that a is not the centre of the town, but a distance out from it, with a < b.

    So I have stated that the number of people living in this area will be the population density multiplied by area. Easy enough. I have then partitioned up the large ring in between a and b into small rings.

    Here's what I have so far:

    [tex]Area_{i} = \pi(r^{2}_{i}-r^{2}_{i-1})[/tex]

    and

    [tex]\Delta r_{i} = r_{i}-r_{i-1}[/tex]

    and it follows that

    [tex]Population \approx \sum^{n}_{i=1}f(r^{i}_{*})\pi(r^{2}_{i}-r^{2}_{i-1})[/tex]

    My question is, how do I convert this to an integral if there's no [tex]\Delta r[/tex] involved?

    I do realize that as the rings get really small they can be approximated as circumferences but this still doesn't help me get the term I need.

    Thanks for any help.
     
  2. jcsd
  3. Aug 18, 2008 #2

    tiny-tim

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    Homework Helper

    Welcome to PF!

    Hi Andrusko! Welcome to PF! :smile:

    (I assume you mean the number of people living between arbitrary distances a and b from the centre of the town? :smile:)

    No, using [tex]\text{Area}_{i} = \pi(r^{2}_{i}-r^{2}_{i-1})[/tex] is just a nuisance, and it doesn't incorporate d-anything.

    You have correctly sliced up the area … in this case, into rings.

    ok, call the thickness of each ring dr …

    then its area (to first order of magnitude) is … ?

    and so the total area is ∫ … dr ? :smile:
     
  4. Aug 18, 2008 #3
    Ah, so I should treat it like a rectangle and go:

    [tex] Area_{i} = 2\pi r^{*}_{i} \Delta r_{i} [/tex]

    Then the Riemann sum becomes:

    [tex]Population \approx \sum^{n}_{i=1} f(r^{*}_{i})2\pi r^{*}_{i} \Delta r_{i}[/tex]

    Which in the limit as [tex]\Delta r_{i} \rightarrow 0[/tex] becomes:

    [tex]Population = \int^{b}_{a} f(r)2\pi rdr[/tex]
    [tex]Population = 2\pi\int^{b}_{a} rf(r)dr[/tex]

    This seems to have solved it. Thankyou for the help.
     
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