Let \(f(x, y)=x^{2}y\) be a function defined over the square having its lower left corner at \((1, 0)\) and its upper right corner at \((5, 4)\), as shown in Figure 2.
Solution (a) We begin by partitioning the square in Figure 2 into \(N=4\) congruent subsquares, as shown in Figure 3. For each subsquare, \(\Delta A_{k}=(2) (2) =4\). The Riemann sum for which \( (u_{k},v_{k})\) is the lower left corner of each subsquare is \[ \begin{eqnarray*} \sum_{k=1}^{4}f(u_{k},v_{k})\Delta A_{k} &=&f(1, 0)(4) +f(3, 0) (4) +f(1, 2) (4) +f(3, 2) (4) \\[3pt] &=&[f(1, 0)+f(3, 0) +f(1, 2) +f(3, 2)] (4)\\[3pt] &=&(0+0+2+18) (4) =80 \qquad\qquad {\color{#0066A7}{\hbox{\(f(x, y) =x^{2}y\)}}} \end{eqnarray*} \]
(b) See Figure 3. The Riemann sum using the upper right corner of each subsquare for \((u_{k},v_{k})\) is \[ \begin{eqnarray*} \sum_{k=1}^{4}f(u_{k},v_{k})\Delta A_{k}& =&f(3, 2) (4) +f(5, 2) (4) +f(3, 4) (4) +f(5, 4) (4) \\[4pt] &=&[ f(3, 2) +f(5, 2) +f(3, 4)+f(5, 4) ] (4)\\[4pt] &=&(18+50+36+100)(4) =816 \end{eqnarray*} \]
(c) Now we partition the square in Figure 2 into \(N=8\) congruent rectangles with sides \(\Delta x_{i}=2,\) \(i=1, 2,\) and \(\Delta y_{j}=1\), \( j=1, 2, 3, 4,\) as shown in Figure 4. The Riemann sum, for which \((u_{k}, v_{k}),\) \(k=1, 2, 3,\ldots, 8,\) is the lower right corner of each rectangle and \( \Delta A_{k}=(2) (1) =2\), is \[ \begin{eqnarray*} \sum_{k=1}^{8}f(u_{k},v_{k})\Delta A_{k} &=&f(3,0) (2) +f(5,0) (2) +f(3,1) (2) +f(5,1) (2) +f(3,2) (2)\\[4pt] &&+\,f(5,2) (2) +f(3,3) (2) +f(5,3) (2) \\[4pt] &=&[ f(3,0) +f(5,0) +f(3,1) +f(5,1) +f(3,2) +f(5,2)\\[4pt] &&+\,f(3,3) +f(5,3) ] (2) \\[4pt] &=&( 0+0+9+25+18+50+27+75) (2) =408 \end{eqnarray*} \]