| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Trapezium rule with reasoning |
| Difficulty | Moderate -0.3 This is a straightforward application of standard techniques: trapezium rule with only 2 strips (minimal computation), explaining why the estimate overestimates (concave down curve - standard observation), and volume of revolution which simplifies nicely since (√(1+e^{4x}))² = 1+e^{4x} integrates easily. All parts are routine textbook exercises requiring no problem-solving insight, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_5}
The diagram shows the curve $y = \sqrt{1 + e^{4x}}$ for $0 \leq x \leq 6$. The region bounded by the curve and the lines $x = 0$, $x = 6$ and $y = 0$ is denoted by $R$.
\begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with 2 strips to find an estimate of the area of $R$, giving your answer correct to 2 decimal places. [3]
\item With reference to the diagram, explain why this estimate is greater than the exact area of $R$. [1]
\item The region $R$ is rotated completely about the $x$-axis. Find the exact volume of the solid produced. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q5 [8]}}