15.
In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable.
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.
(2)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-30_627_581_1142_404}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-30_524_442_1238_1183}
\captionsetup{labelformat=empty}
\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.