Answer
(a) 288
(b) 144
Work Step by Step
Given $R=\{(x, y) | 0 \leqslant x \leqslant 6,0 \leqslant y \leqslant 4\}$, $z=f(x,y)=xy$ and $\Delta A=4$
(a) By using the Riemann sum
\begin{align*}
V& \approx \sum_{i=1}^{3} \sum_{j=1}^{2} f\left(x_{i}, y_{j}\right) \Delta A\\
&=[f(2,2)+f(2,4) +f(4,2) +f(4,4) +f(6,2) +f(6,4)] \Delta A\\
&=[4 +8 +8 +16 +12 +24 ](4)\\
&=288
\end{align*}
(b) By using Midpoint Rule
$$V\approx \sum_{i=1}^{m} \sum_{j=1}^{n} f\left(\overline{X}_{i}, \overline{y}_{j}\right) \Delta A$$
where $\overline{x}_{i}$ is the midpoint of $\left[x_{i-1}, x_{i}\right]$ and $\overline{y}_{j}$ is the midpoint of $\left[y_{j-1}, y_{j}\right] .$
\begin{align*}
V& \approx \sum_{i=1}^{3} \sum_{j=1}^{2} f\left(\overline{x}_{i}, \overline{y}_{j}\right) \Delta A\\
&=[f(1,1)+f(1,3)+f(3,1) +f(3,3) +f(5,1) +f(5,3)] \Delta A\\
&=[1+3+3+9+5+15 ](4)\\
&=144
\end{align*}
