Answer
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} f\left( {x,y} \right){\rm{d}}A \approx 57.01$
Work Step by Step
Using Eq. (11) and the values in the given table, we estimate $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} f\left( {x,y} \right){\rm{d}}A$:
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} f\left( {x,y} \right){\rm{d}}A \approx \mathop \sum \limits_{j = 1}^6 f\left( {{P_j}} \right)Area\left( {{{\cal D}_j}} \right)$
$ = 9\cdot1.2 + 9.1\cdot1.1 + 9.3\cdot1.4 + 9.1\cdot0.6 + 8.9\cdot1.2 + 8.8\cdot0.8$
$ = 57.01$
Thus, $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} f\left( {x,y} \right){\rm{d}}A \approx 57.01$.
