Moderate -0.8 This is a straightforward application of the trapezium rule formula with clearly specified parameters (4 strips, simple interval). Part (i) requires only routine calculation with a calculator, while part (ii) tests basic understanding of concavity and trapezium rule over/underestimation. Both parts are standard textbook exercises requiring no problem-solving insight.
Use the trapezium rule with four strips to estimate \(\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing your working.
Fig. 1 shows a sketch of \(y = \sqrt { 1 + \mathrm { e } ^ { x } }\).
\begin{figure}[h]
Suppose that the trapezium rule is used with more strips than in part (i) to estimate \(\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x\). State, with a reason but no further calculation, whether this would give a larger or smaller estimate.
1 (i) Use the trapezium rule with four strips to estimate $\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x$, showing your working.
Fig. 1 shows a sketch of $y = \sqrt { 1 + \mathrm { e } ^ { x } }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f657e167-e6f8-4df2-901b-067c32835877-02_535_1074_571_532}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
(ii) Suppose that the trapezium rule is used with more strips than in part (i) to estimate $\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x$. State, with a reason but no further calculation, whether this would give a larger or smaller estimate.
\hfill \mbox{\textit{OCR MEI C4 2011 Q1 [6]}}