Answer
Choice 1. Use midpoints as sample points (Figure 16 (A))
${S_{3,2}} = 42$
Choice 2. Use the sample points as in Figure 16 (B)
${S_{3,2}} = 42$
Work Step by Step
Compute the Riemann sums for the double integral $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} f\left( {x,y} \right){\rm{d}}A$, where ${\cal R} = \left[ {1,4} \right] \times \left[ {1,3} \right]$.
From Figure 16, we obtain the grid: $N \times M = 3 \times 2$.
Using the regular partition, we get the dimensions of the subrectangles:
$\Delta x = \frac{{4 - 1}}{3} = 1$, ${\ \ \ \ }$ $\Delta y = \frac{{3 - 1}}{2} = 1$
The Riemann sum ${S_{3,2}}$ is given by
${S_{3,2}} = \mathop \sum \limits_{i = 1}^3 \mathop \sum \limits_{j = 1}^2 f\left( {{P_{ij}}} \right)\Delta {x_i}\Delta {y_j} = \mathop \sum \limits_{i = 1}^3 \mathop \sum \limits_{j = 1}^2 f\left( {{P_{ij}}} \right)$
Choice 1. Use midpoints as sample points (Figure 16 (A))
${S_{3,2}} = f\left( {\frac{3}{2},\frac{3}{2}} \right) + f\left( {\frac{5}{2},\frac{3}{2}} \right) + f\left( {\frac{7}{2},\frac{3}{2}} \right) + f\left( {\frac{3}{2},\frac{5}{2}} \right) + f\left( {\frac{5}{2},\frac{5}{2}} \right) + f\left( {\frac{7}{2},\frac{5}{2}} \right)$
$ = 7 + 7 + 7 + 7 + 7 + 7$
${S_{3,2}} = 42$
Choice 2. Use the sample points as in Figure 16 (B)
The sample points are: $\left( {\frac{3}{2},\frac{3}{2}} \right)$, $\left( {2,1} \right)$, $\left( {\frac{7}{2},\frac{3}{2}} \right)$, $\left( {2,3} \right)$, $\left( {\frac{5}{2},\frac{5}{2}} \right)$, $\left( {4,3} \right)$.
So,
${S_{3,2}} = f\left( {\frac{3}{2},\frac{3}{2}} \right) + f\left( {2,1} \right) + f\left( {\frac{7}{2},\frac{3}{2}} \right) + f\left( {2,3} \right) + f\left( {\frac{5}{2},\frac{5}{2}} \right) + f\left( {4,3} \right)$
$ = 7 + 7 + 7 + 7 + 7 + 7$
${S_{3,2}} = 42$
