| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Emblem or applied region area |
| Difficulty | Standard +0.8 This C4 question combines parametric equations with integration to find area, requiring students to set up and evaluate ∫y dx/dt dt with trigonometric substitution. The 7-mark part (b) involves multiple steps including finding dx/dt, setting up the integral with correct limits, and integrating products of trigonometric functions—more demanding than typical C4 area questions but still within standard syllabus techniques. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.08d Evaluate definite integrals: between limits |
\includegraphics{figure_1}
Figure 1 shows a cross-section $R$ of a dam. The line $AC$ is the vertical face of the dam, $AB$ is the horizontal base and the curve $BC$ is the profile. Taking $x$ and $y$ to be the horizontal and vertical axes, then $A$, $B$ and $C$ have coordinates $(0, 0)$, $(3\pi^2, 0)$ and $(0, 30)$ respectively. The area of the cross-section is to be calculated.
Initially the profile $BC$ is approximated by a straight line.
\begin{enumerate}[label=(\alph*)]
\item Find an estimate for the area of the cross-section $R$ using this approximation. [1]
\end{enumerate}
The profile $BC$ is actually described by the parametric equations.
$$x = 16t^2 - \pi^2, \quad y = 30 \sin 2t, \quad \frac{\pi}{4} \leq t \leq \frac{\pi}{2}.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the exact area of the cross-section $R$. [7]
\item Calculate the percentage error in the estimate of the area of the cross-section $R$ that you found in part (a). [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q4 [10]}}