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Math Help - population of a city

  1. #1
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    Exclamation population of a city

    Hi there
    my question is

    a city is laid out in the shape of a circle with a radius of 10 km.
    there is a motorway that runs from east to west of the city along the centre.

    The population density at a distance y km North or South is given by the function (D)= 4y
    (so most people tend to live away from the motorway).

    What is the total population of the city(or how would u find it)?

    Thanks a lot
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  2. #2
    MHF Contributor chisigma's Avatar
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    Is...

    \displaystyle u= \int_{C} 4 y\ dx\ dy (1)

    ... where C is the circle 'centered' in the origin and with radius 10. In polar coordinates the (1) is...

    \displaystyle u= 4\ \int_{0}^{10} \int_{-\pi}^{\pi} r^{2}\ \sin \theta\ d r\ d \theta (2)

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor chisigma's Avatar
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    There is a little detail... the density cannot be negative so that is D=4\ |y| and the integral becomes...

    \displaystyle u= 8\ \int_{0}^{10} \int_{0}^{\pi} r^{2}\ \sin \theta\ d r\ d \theta (1)

    Kind regards

    \chi \sigma
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