14 Fig. 14.1 shows the curve with equation \(y = \frac { 1 } { 1 + x ^ { 2 } }\), together with 5 rectangles of equal width.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57007d39-abb0-475e-9ed8-03021fa1273b-10_940_1557_331_246}
\captionsetup{labelformat=empty}
\caption{Fig. 14.1}
\end{figure}
Fig. 14.2 shows the coordinates of the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
| Point | A | B | C | D | E | F |
| \(x\) | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
| \(y\) | 1 | 0.96154 | 0.86207 | 0.73529 | 0.60976 | 0.5 |
\section*{Fig. 14.2}
- Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to show that a lower bound for \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm { dx }\) is 0.7337, correct to \(\mathbf { 4 }\) decimal places.
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[2] - Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to calculate an upper bound for \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm {~d} x\) correct to \(\mathbf { 4 }\) decimal places.
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[2] - Hence find the length of the interval in which your answers to parts (a) and (b) indicate the value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm {~d} x\) lies.
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[1]
Amit uses \(n\) rectangles, each of width \(\frac { 1 } { n }\), to calculate upper and lower bounds for \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm {~d} x\), using different values of \(n\). His results are shown in Fig. 14.3.
| \(n\) | 10 | 20 | 40 |
| upper bound | 0.80998 | 0.79779 | 0.79162 |
| lower bound | 0.75998 | 0.77279 | 0.77912 |
- Find the length of the smallest interval in which Amit now knows \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } \mathrm { dx }\) lies.
- Without doing any calculation, explain how Amit could find a smaller interval which contains the value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { 2 } } d x\).