| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 15 |
| Topic | Volumes of Revolution |
| Type | Surface area of revolution: Cartesian curve |
| Difficulty | Challenging +1.8 This is a Pre-U Further Maths question requiring arc length and surface of revolution formulas with integration. Part (i) involves computing √(1+(dy/dx)²) which simplifies nicely to (x²+1)/(2x), leading to a tractable integral. Part (ii) requires the surface area formula with careful algebraic manipulation and integration by parts to reach the exact form. While the techniques are standard for Further Maths, the multi-step integration, algebraic simplification to exact forms, and the 10-mark allocation indicate substantial computational demand beyond typical A-level questions. |
| Spec | 4.08d Volumes of revolution: about x and y axes4.08e Mean value of function: using integral |
The curve $C$ is given by $y = \frac{1}{4}x^2 - \frac{1}{2}\ln x$ for $2 \leq x \leq 8$.
\begin{enumerate}[label=(\roman*)]
\item Find, in its simplest exact form, the length of $C$. [5]
\item When $C$ is rotated through $2\pi$ radians about the $x$-axis, a surface of revolution is formed. Show that the area of this surface is $\pi(270 - 47\ln 2 - 2(\ln 2)^2)$. [10]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2018 Q12 [15]}}