| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Standard +0.3 Part (a) requires understanding that the trapezium rule overestimates for convex curves (1 mark of conceptual recall). Part (b) is algebraic manipulation: recognizing that the new integrand equals (2x² - x) + (1 + 3x), then using linearity of integration to add the trapezium estimate to an easily computed integral of a linear function. This is slightly easier than average as it tests standard integration properties with minimal calculation, though the manipulation requires some insight. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
\includegraphics{figure_3}
Figure 3 shows a sketch of the curve with equation $y = 2x^2 - x$.
The finite region $R$, shown shaded in Figure 3, is bounded by the curve, the line with equation $x = -0.5$, the $x$-axis and the line with equation $x = 1.5$.
\begin{enumerate}[label=(\alph*)]
\item The trapezium rule with four strips is used to find an estimate for the area of $R$. Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of $R$. [1]
\end{enumerate}
The estimate for R is found to be 2.58.
Using this value, and showing your working,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item estimate the value of $\int_{-0.5}^{1.5} (2x^2 + 1 + 2x) \, dx$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2022 Q17 [4]}}