Answer
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S \approx \frac{1}{2}\left( {\sqrt 5 + \sqrt {10} } \right)$
Work Step by Step
From Figure 17 we see that the domain ${\cal D}$ is given by ${\cal D} = \left\{ {\left( {u,v} \right):0 \le u \le 1,0 \le v \le 1} \right\}$.
Using $f\left( {G\left( {u,v} \right)} \right) = u + v$, we obtain
$f\left( A \right) = \frac{1}{4} + \frac{3}{4} = 1$, ${\ \ \ \ }$ $f\left( B \right) = \frac{3}{4} + \frac{3}{4} = \frac{3}{2}$
$f\left( C \right) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$, ${\ \ \ \ }$ $f\left( D \right) = \frac{3}{4} + \frac{1}{4} = 1$
From ${\bf{N}}\left( A \right) = \left( {2,1,0} \right)$, ${\bf{N}}\left( B \right) = \left( {1,3,0} \right)$, ${\bf{N}}\left( C \right) = \left( {3,0,1} \right)$, and ${\bf{N}}\left( D \right) = \left( {2,0,1} \right)$, we get
$||{\bf{N}}\left( A \right)|| = \sqrt {{2^2} + {1^2} + 0} = \sqrt 5 $
$||{\bf{N}}\left( B \right)|| = \sqrt {{1^2} + {3^2} + 0} = \sqrt {10} $
$||{\bf{N}}\left( C \right)|| = \sqrt {{3^2} + 0 + {1^2}} = \sqrt {10} $
$||{\bf{N}}\left( D \right)|| = \sqrt {{2^2} + 0 + {1^2}} = \sqrt 5 $
Using the Riemann sum we estimate $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S$ by
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S \approx \mathop \sum \limits_{i = 1}^4 {f_i}\left( {G\left( {u,v} \right)} \right)||{{\bf{N}}_i}\left( {u,v} \right)||\Delta u\Delta v$
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S \approx $
$f\left( A \right)||{\bf{N}}\left( A \right)||\Delta u\Delta v + f\left( B \right)||{\bf{N}}\left( B \right)||\Delta u\Delta v$
$ + f\left( C \right)||{\bf{N}}\left( C \right)||\Delta u\Delta v + f\left( D \right)||{\bf{N}}\left( D \right)||\Delta u\Delta v$
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S \approx $
$\sqrt 5 \Delta u\Delta v + \frac{3}{2}\sqrt {10} \Delta u\Delta v + \frac{1}{2}\sqrt {10} \Delta u\Delta v + \sqrt 5 \Delta u\Delta v$
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S \approx \left( {\sqrt 5 + \frac{3}{2}\sqrt {10} + \frac{1}{2}\sqrt {10} + \sqrt 5 } \right)\Delta u\Delta v$
From Figure 17 we see that $\Delta u = \frac{1}{2}$ and $\Delta v = \frac{1}{2}$, so
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S \approx \left( {2\sqrt 5 + 2\sqrt {10} } \right)\frac{1}{4} = \frac{1}{2}\left( {\sqrt 5 + \sqrt {10} } \right)$
We get
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} f\left( {x,y,z} \right){\rm{d}}S \approx \frac{1}{2}\left( {\sqrt 5 + \sqrt {10} } \right)$.