Answer
The total mass is $\frac{{15}}{{64}}$ kg.
Work Step by Step
We have a mass density of $\delta \left( {x,y} \right) = \frac{y}{x}$ and the region $1 \le x \le 4$, $0 \le y \le {x^{ - 1}}$.
Using Eq. (1), the total mass is given by
${\rm{total{\ }mass}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \delta \left( {x,y} \right){\rm{d}}A = \mathop \smallint \limits_{x = 1}^4 \mathop \smallint \limits_{y = 0}^{{x^{ - 1}}} \frac{y}{x}{\rm{d}}y{\rm{d}}x$
$ = \frac{1}{2}\mathop \smallint \limits_{x = 1}^4 \left( {\left( {\frac{{{y^2}}}{x}} \right)|_0^{{x^{ - 1}}}} \right){\rm{d}}x$
$ = \frac{1}{2}\mathop \smallint \limits_{x = 1}^4 \left( {{x^{ - 3}}} \right){\rm{d}}x$
$ = - \frac{1}{4}\left( {{x^{ - 2}}|_1^4} \right) = - \frac{1}{4}\left( {\frac{1}{{16}} - 1} \right) = \frac{{15}}{{64}}$
The total mass is $\frac{{15}}{{64}}$ kg.
