Answer
The total population is $1912$ people.
Work Step by Step
We have a population density of $\delta \left( {x,y} \right) = 100{{\rm{e}}^{ - 0.1y}}$ and the region within the sector $2\left| x \right| \le y \le 8$.
Write the sector:
$\left\{ {\begin{array}{*{20}{c}}
{2x \le y \le 8}&{{\rm{for}{\ }}x \ge 0}\\
{ - 2x \le y \le 8}&{{\rm{for}{\ }}x < 0}
\end{array}} \right.$
We sketch the region ${\cal D}$ and notice that it is bounded below by $y=0$ and bounded above by the line $y=8$. Whereas, it is bounded left by the line $y=-2x$ and bounded right by the line $y=2x$. So, we can consider ${\cal D}$ as a horizontally simple region with the description:
${\cal D} = \left\{ {\left( {x,y} \right)|0 \le y \le 8, - \frac{y}{2} \le x \le \frac{y}{2}} \right\}$
Using Eq. (1), the total population is given by
${\rm{total{\ }population}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \delta \left( {x,y} \right){\rm{d}}A$
$ = \mathop \smallint \limits_{y = 0}^8 \mathop \smallint \limits_{x = - y/2}^{y/2} 100{{\rm{e}}^{ - 0.1y}}{\rm{d}}x{\rm{d}}y$
$ = 100\mathop \smallint \limits_{y = 0}^8 {{\rm{e}}^{ - 0.1y}}\left( {x|_{ - y/2}^{y/2}} \right){\rm{d}}y$
$ = 100\mathop \smallint \limits_{y = 0}^8 y{{\rm{e}}^{ - 0.1y}}{\rm{d}}y$
Write $u=y$ and $dv = {{\rm{e}}^{ - 0.1y}}dy$. So, $du = dy$ and $v = - 10{{\rm{e}}^{ - 0.1y}}$. Using Integration by Parts Formula (Section 8.1),
$\smallint u{\rm{d}}v = uv - \smallint v{\rm{d}}u$
we get
${\rm{total{\ }population}} = 100\mathop \smallint \limits_{y = 0}^8 y{{\rm{e}}^{ - 0.1y}}{\rm{d}}y$
$ = 100\left( {\left( { - 10y{{\rm{e}}^{ - 0.1y}}} \right)|_0^8 + 10\mathop \smallint \limits_{y = 0}^8 {{\rm{e}}^{ - 0.1y}}{\rm{d}}y} \right)$
$ = 100\left( { - 80{{\rm{e}}^{ - 0.8}} - 100\left( {{{\rm{e}}^{ - 0.1y}}|_0^8} \right)} \right)$
$ = 100\left( { - 80{{\rm{e}}^{ - 0.8}} - 100\left( {{{\rm{e}}^{ - 0.8}} - 1} \right)} \right)$
$ = 100\left( { - 80{{\rm{e}}^{ - 0.8}} - 100{{\rm{e}}^{ - 0.8}} + 100} \right)$
$ = 100\left( {100 - 180{{\rm{e}}^{ - 0.8}}} \right) \simeq 1912.08$
So, the total population is $1912$ people.
