| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Topic | Areas by integration |
| Type | Trapezium rule estimation |
| Difficulty | Standard +0.3 This is a straightforward two-part question combining trapezium rule (routine A-level technique) with finding area between curves. Part (a) is mechanical application of a formula, part (b) requires subtracting one integral from another and calculating a percentage. The calculus in part (b) is simple (integrating a quadratic), and the question clearly guides students through the steps. Slightly easier than average due to its structured nature and standard techniques. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 3 | 2.6833 | 2.4 | 2.1466 | 1.92 |
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.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline
$y$ & 3 & 2.6833 & 2.4 & 2.1466 & 1.92 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item 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]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-30_627_581_1142_404}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{b063f4ea-372b-4193-b8fe-a9f8017d7349-30_524_442_1238_1183}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
The region $R$, shown shaded in Figure 1, is bounded by
\begin{itemize}
\item the curve $C _ { 1 }$
\item the curve $C _ { 2 }$ with equation $y = 2 - \frac { 1 } { 4 } x ^ { 2 }$
\item the line with equation $x = 2$
\item the $y$-axis
\end{itemize}
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),
\item calculate an estimate for the percentage of the logo which is shaded.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q15 [5]}}