6 Alex is investigating the area, \(A\), under the graph of \(y = x ^ { 2 }\) between \(x = 1\) and \(x = 1.5\). They draw the graph, together with rectangles of width \(\delta x = 0.1\), and varying heights \(y\).
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-06_531_714_356_251}
- Use the rectangles in the diagram to show that lower and upper bounds for the area \(A\) are 0.73 and 0.855 respectively.
- Alex finds lower and upper bounds for the area \(A\), using widths \(\delta x\) of decreasing size. The results are shown in the table. Where relevant, values are given correct to 3 significant figures.
| Width \(\delta x\) | 0.1 | 0.05 | 0.025 | 0.0125 |
| Lower bound for area \(A\) | 0.73 | 0.761 | 0.776 | 0.784 |
| Upper bound for area \(A\) | 0.855 | 0.823 | 0.807 | 0.799 |
- Write down an expression, in terms of \(y\) and \(\delta x\), for the exact value of the area \(A\).