Answer
The total population is $72175$ people.
Work Step by Step
We have a population density of $\delta \left( {x,y} \right) = 2000{\left( {{x^2} + {y^2}} \right)^{ - 0.2}}$ and the region within a $4$-km radius, that is ${x^2} + {y^2} \le 16$.
Using Eq. (1), the total population is given by
${\rm{total{\ }population}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \delta \left( {x,y} \right){\rm{d}}A$
In polar coordinates, the region description of ${\cal D}$ is
${\cal D} = \left\{ {\left( {r,\theta } \right)|0 \le r \le 4,0 \le \theta \le 2\pi } \right\}$
Using polar coordinates, we get
${\rm{total{\ }population}} = \mathop \smallint \limits_{\theta = 0}^{2\pi } \mathop \smallint \limits_{r = 0}^4 \delta \left( {r\cos \theta ,r\sin \theta } \right)r{\rm{d}}r{\rm{d}}\theta $
$ = \mathop \smallint \limits_{\theta = 0}^{2\pi } \mathop \smallint \limits_{r = 0}^4 2000{r^{ - 0.4}}r{\rm{d}}r{\rm{d}}\theta $
$ = \mathop \smallint \limits_{\theta = 0}^{2\pi } \mathop \smallint \limits_{r = 0}^4 2000{r^{0.6}}{\rm{d}}r{\rm{d}}\theta $
$ = \frac{{2000}}{{1.6}}\mathop \smallint \limits_{\theta = 0}^{2\pi } \left( {{r^{1.6}}|_0^4} \right){\rm{d}}\theta $
$ = \frac{{2000 \times {4^{1.6}}}}{{1.6}}\mathop \smallint \limits_{\theta = 0}^{2\pi } {\rm{d}}\theta = \frac{{4000\pi \times {4^{1.6}}}}{{1.6}} \simeq 72174.8$
The total population is $72175$ people.
