2022 YIJC P1 Q2

Figure 1 shows a sketch of the curve \(y=\mathrm{f}\left( x \right)\) and \(A\) is the region under the curve between \(x=1\) and \(x=3\). Yvonne and Irene use different ways to draw \(4\) rectangles of equal width, \(h\), to estimate the area of \(A\).

Figure 2 shows \(4\) rectangles drawn by Yvonne, with the curve intersecting each rectangle at the mid-point of its width. The \(x\)-coordinates of the mid-points are \({{m}_{1}}\), \({{m}_{2}}\), \({{m}_{3}}\) and \({{m}_{4}}\).

1. State the values of \(h\), \({{m}_{1}}\), \({{m}_{2}}\), \({{m}_{3}}\) and \({{m}_{4}}\).[2]
2. The sum of area of rectangles using Yvonne’s method is denoted by \(B\). Show that \(B=h\sum\limits_{r=0}^{3}{\left( \mathrm{f}\left( a+rh \right) \right)}\), where the value of \(a\) is to be determined.[2]
3. Irene finds that the sum of area of \(4\) rectangles that she has drawn is \(C=h\sum\limits_{r=0}^{3}{\left( \mathrm{f}\left( 1+rh \right) \right)}\). Draw these rectangles in Figure 1.[1]

You are now given \(\mathrm{f}\left( x \right)=\frac{1}{x}+1\).

4. By finding the numerical values of \(B\), \(C\) and the actual area of region \(A\), explain how these values and the rectangles drawn in Figures 1 and 2 show that Yvonne’s estimation of the area of \(A\) is better than Irene’s.[2]