Draw a picture of \(R\text{.}\) Partition \([0,2]\) into 2 subintervals of equal length and the interval \([1,3]\) into two subintervals of equal length. Draw these partitions on your picture of \(R\) and label the resulting subrectangles using the labeling scheme we established in the definition of a double Riemann sum.
For each \(i\) and \(j\text{,}\) let \((x_{ij}^*, y_{ij}^*)\) be the midpoint of the rectangle \(R_{ij}\text{.}\) Identify the coordinates of each \((x_{ij}^*, y_{ij}^*)\text{.}\) Draw these points on your picture of \(R\text{.}\)
\begin{equation*}
\sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A
\end{equation*}
using the partitions we have described. If we let \((x_{ij}^*, y_{ij}^*)\) be the midpoint of the rectangle \(R_{ij}\) for each \(i\) and \(j\text{,}\) then the resulting Riemann sum is called a midpoint sum.
