
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

Is...
$\displaystyle \displaystyle u= \int_{C} 4 y\ dx\ dy$ (1)
... where C is the circle 'centered' in the origin and with radius 10. In polar coordinates the (1) is...
$\displaystyle \displaystyle u= 4\ \int_{0}^{10} \int_{\pi}^{\pi} r^{2}\ \sin \theta\ d r\ d \theta$ (2)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$

There is a little detail... the density cannot be negative so that is $\displaystyle D=4\ y$ and the integral becomes...
$\displaystyle \displaystyle u= 8\ \int_{0}^{10} \int_{0}^{\pi} r^{2}\ \sin \theta\ d r\ d \theta$ (1)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$