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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.

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# Homework Help: Integral involving population density

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