Answer
We choose $N=4$ and $M=2$. So, the grid has eight subrectangles.
We compute two different Riemann sums to estimate $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} \left( {x + y} \right){\rm{d}}A$ using:
1. lower-left vertices as sample points
${S_{4,2}} = 0.625$
2. midpoints as sample points
${S_{4,2}} = 1$
Work Step by Step
We have $f\left( {x,y} \right) = x + y$ and ${\cal R} = \left[ {0,1} \right] \times \left[ {0,1} \right]$.
We choose $N=4$ and $M=2$. So, the grid has eight subrectangles.
Using the regular partition, we get the dimensions of the subrectangles:
$\Delta x = \frac{{1 - 0}}{4} = \frac{1}{4}$, ${\ \ \ \ }$ $\Delta y = \frac{{1 - 0}}{2} = \frac{1}{2}$
The Riemann sum ${S_{4,2}}$ to estimate $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal R}^{} \left( {x + y} \right){\rm{d}}A$ is given by
${S_{4,2}} = \mathop \sum \limits_{i = 1}^4 \mathop \sum \limits_{j = 1}^2 f\left( {{P_{ij}}} \right)\Delta {x_i}\Delta {y_j} = \frac{1}{8}\mathop \sum \limits_{i = 1}^4 \mathop \sum \limits_{j = 1}^2 f\left( {{P_{ij}}} \right)$
Method 1. Use lower-left vertices as sample points
${S_{4,2}} = \frac{1}{8}(f\left( {0,0} \right) + f\left( {\frac{1}{4},0} \right) + f\left( {\frac{1}{2},0} \right) + f\left( {\frac{3}{4},0} \right)$
${\ \ \ \ \ \ \ \ \ \ \ }$ $ + f\left( {0,\frac{1}{2}} \right) + f\left( {\frac{1}{4},\frac{1}{2}} \right) + f\left( {\frac{1}{2},\frac{1}{2}} \right) + f\left( {\frac{3}{4},\frac{1}{2}} \right))$
$ = \frac{1}{8}\left( {0 + \frac{1}{4} + \frac{1}{2} + \frac{3}{4} + \frac{1}{2} + \frac{3}{4} + 1 + \frac{5}{4}} \right)$
${S_{4,2}} = 0.625$
Method 2. Use midpoints as sample points
${S_{4,2}} = \frac{1}{8}(f\left( {\frac{1}{8},\frac{1}{4}} \right) + f\left( {\frac{3}{8},\frac{1}{4}} \right) + f\left( {\frac{5}{8},\frac{1}{4}} \right) + f\left( {\frac{7}{8},\frac{1}{4}} \right)$
${\ \ \ \ \ \ \ \ \ \ \ }$ $ + f\left( {\frac{1}{8},\frac{3}{4}} \right) + f\left( {\frac{3}{8},\frac{3}{4}} \right) + f\left( {\frac{5}{8},\frac{3}{4}} \right) + f\left( {\frac{7}{8},\frac{3}{4}} \right))$
$ = \frac{1}{8}\left( {\frac{3}{8} + \frac{5}{8} + \frac{7}{8} + \frac{9}{8} + \frac{7}{8} + \frac{9}{8} + \frac{{11}}{8} + \frac{{13}}{8}} \right)$
${S_{4,2}} = 1$
