Finding Riemann Sums

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.

  1. Find a Riemann sum of \(f\) over this region by partitioning the square into four congruent subsquares. Choose the lower left corner of each subsquare as \((u_{k},v_{k})\), \(k=1, 2, 3, 4.\)
  2. Find a Riemann sum of \(f\) over the partition used in (a) but choose the upper right corner of each subsquare as \((u_{k},v_{k})\), \(k=1, 2, 3, 4.\)
  3. Find a Riemann sum of \(f\) over this region by partitioning the square into eight congruent rectangles with sides of length \(\Delta x_{i}=2,\) \(i=1, 2,\) and \(\Delta y_{j}=1,\) \(j=1, 2, 3, 4\). Choose the lower right corner of each rectangle as \((u_{k}, v_{k}) ,\) \(k=1, 2, \ldots, 8.\)

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 \[ \begin{eqnarray*} \sum_{k=1}^{8}f(u_{k},v_{k})\Delta A_{k} &=&f(3,0) (2) +f(5,0) (2) +f(3,1) (2) +f(5,1) (2) +f(3,2) (2)\\[4pt] &&+\,f(5,2) (2) +f(3,3) (2) +f(5,3) (2) \\[4pt] &=&[ f(3,0) +f(5,0) +f(3,1) +f(5,1) +f(3,2) +f(5,2)\\[4pt] &&+\,f(3,3) +f(5,3) ] (2) \\[4pt] &=&( 0+0+9+25+18+50+27+75) (2) =408 \end{eqnarray*} \]