| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Topic | Numerical integration |
| Type | Apply trapezium rule to given table |
| Difficulty | Moderate -0.3 Part (a) is a standard trapezium rule application with given y-values requiring only substitution into the formula. Parts (b)(i) and (b)(ii) test understanding of integral properties (adding constants, translation) but are routine manipulations once part (a) is complete. This is slightly easier than average due to minimal calculation complexity and straightforward conceptual steps. |
| Spec | 1.08h Integration by substitution1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 0.5 | 1 | 1.5 | 2 |
| \(y\) | 1 | \(e^{0.05}\) | \(e^{0.2}\) | \(e^{0.45}\) | \(e^{0.8}\) |
\includegraphics{figure_1}
Figure 1 shows part of the curve with equation $y = e^{\frac{1}{5}x^2}$ for $x \geq 0$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $y$-axis, the $x$-axis, and the line with equation $x = 2$
The table below shows corresponding values of $x$ and $y$ for $y = e^{\frac{1}{5}x^2}$
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline
$y$ & 1 & $e^{0.05}$ & $e^{0.2}$ & $e^{0.45}$ & $e^{0.8}$ \\
\hline
\end{tabular}
(a) Use the trapezium rule, with all the values of $y$ in the table, to find an estimate for the area of $R$, giving your answer to 2 decimal places. [3]
(b) Use your answer to part (a) to deduce an estimate for
(i) $\int_0^2 \left( 4 + e^{\frac{1}{5}x^2} \right) dx$
(ii) $\int_1^3 e^{\frac{1}{5}(x-1)^2} dx$
giving your answers to 2 decimal places. [2]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q5 [5]}}