| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 8 |
| Topic | Volumes of Revolution |
| Type | Volume requiring substitution or integration by parts |
| Difficulty | Challenging +1.2 This is a solid of revolution problem requiring integration of π·y² with a substitution to handle the cube root term. Part (i) involves standard technique with algebraic manipulation (5 marks suggests moderate complexity). Part (ii) requires translating geometric constraints into inequalities for p, then optimizing - this adds problem-solving beyond routine integration. The algebra is non-trivial but follows established A-level methods, placing it moderately above average difficulty. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
The region $R$ between the $x$-axis, the curve $y = \frac{1}{\sqrt{p + x^3}}$ and the lines $x = \sqrt{p}$ and $x = \sqrt{3p}$, where $p$ is a positive parameter, is rotated by $2\pi$ radians about the $x$-axis to form a solid of revolution $S$.
\begin{enumerate}[label=(\roman*)]
\item Find and simplify an algebraic expression, in terms of $p$, for the exact volume of $S$. [5]
\item Given that $R$ must lie entirely between the lines $x = 1$ and $x = \sqrt{48}$ find in exact form
\begin{itemize}
\item the greatest possible value of the volume of $S$
\item the least possible value of the volume of $S$. [3]
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q7 [8]}}