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