- The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).
A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below, with the \(y\) values rounded to 4 decimal places where appropriate.
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 3 | 2.6833 | 2.4 | 2.1466 | 1.92 |
- Use the trapezium rule with all the values of \(y\) in the table to find an approximation for
$$\int _ { 0 } ^ { 2 } f ( x ) d x$$
giving your answer to 3 decimal places.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_629_592_1105_402}
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\caption{Figure 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_540_456_1194_1192}
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\caption{Figure 2}
\end{figure}
The region \(R\), shown shaded in Figure 1, is bounded by
- the curve \(C _ { 1 }\)
- the curve \(C _ { 2 }\) with equation \(y = 2 - \frac { 1 } { 4 } x ^ { 2 }\)
- the line with equation \(x = 2\)
- the \(y\)-axis
The region \(R\) forms part of the design for a logo shown in Figure 2.
The design consists of the shaded region \(R\) inside a rectangle of width 2 and height 3 Using calculus and the answer to part (a), - calculate an estimate for the percentage of the logo which is shaded.