population of a city

**yaz2** · January 16th 2011, 07:59 AM

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

**chisigma** · January 16th 2011, 08:12 AM

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

$\chi$ $\sigma$

**chisigma** · January 17th 2011, 09:32 AM

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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