| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 6 |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Moderate -0.8 Part (a) is a standard trapezium rule application with given values requiring straightforward substitution. Part (b) tests basic integral properties (linearity) to deduce a related integral value—this is routine A-level manipulation requiring only one additional step beyond part (a). The question involves no problem-solving or conceptual challenge beyond textbook exercises. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
| \(y\) | 1.632 | 1.711 | 1.786 | 1.859 | 1.930 |
3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ede204ac-09c3-486b-8877-df935e6ed015-06_709_1052_287_552}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$\\
The table below shows corresponding values of $x$ and $y$ for this curve between $x = 0.5$ and $x = 0.9$
The values of $y$ are given to 4 significant figures.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\
\hline
$y$ & 1.632 & 1.711 & 1.786 & 1.859 & 1.930 \\
\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 estimate for
$$\int _ { 0.5 } ^ { 0.9 } \mathrm { f } ( x ) \mathrm { d } x$$
Give your answer to 3 significant figures.
\item Using your answer to part (a), deduce an estimate for
$$\int _ { 0.5 } ^ { 0.9 } ( 3 \mathrm { f } ( x ) + 2 ) \mathrm { d } x$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q3 [6]}}