Answer
The estimate of $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} f\left( {x,y} \right){\rm{d}}x{\rm{d}}y$ is
${S_{5,4}} = 3.25$
Work Step by Step
From Figure 22, we have $\Delta {x_j} = \frac{{1 - \left( { - 1.5} \right)}}{5} = 0.5$ and $\Delta {y_j} = \frac{{1 - 0}}{4} = 0.25$.
So, $\Delta {A_j} = \Delta {x_j}\cdot\Delta {y_j} = 0.125$.
Using Figure 22, we list the values in the following table:
$\begin{array}{*{20}{c}}
j&1&2&3&4&5&6&7&8&9\\
{f\left( {{P_j}} \right)}&{2.5}&2&3&{2.3}&{2.9}&{3.5}&{3.3}&3&{3.5}
\end{array}$
The Riemann sum is
${S_{5,4}} = \mathop \sum \limits_9^{j = 1} f\left( {{P_j}} \right)\Delta {A_j}$
$ = 0.125\left( {2.5 + 2 + 3 + 2.3 + 2.9 + 3.5 + 3.3 + 3 + 3.5} \right)$
$ = 3.25$
