| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2026 |
| Session | November |
| Marks | 12 |
| Topic | Areas by integration |
| Type | Area under transcendental or composite curve |
| Difficulty | Challenging +1.3 This is a two-part integration question requiring substitution techniques. Part (a) needs recognition that the integrand's derivative appears in the numerator (leading to ln|√(x²+9)|), which is a standard Further Maths technique but requires careful algebraic manipulation. Part (b) involves volume of revolution with a rational function that simplifies nicely. Both parts demand systematic working and algebraic skill, placing this above average difficulty but within reach of well-prepared FM students using standard methods. |
| Spec | 1.08e Area between curve and x-axis: using definite integrals4.08d Volumes of revolution: about x and y axes |
In this question you must show detailed reasoning.
The diagram shows the curve with equation $y = \frac{x + 3}{\sqrt{x^2 + 9}}$.
\includegraphics{figure_8}
The region R, shown shaded in the diagram, is bounded by the curve, the $x$-axis, the $y$-axis, and the line $x = 4$.
\begin{enumerate}[label=(\alph*)]
\item Determine the area of R. Give your answer in the form $p + \ln q$ where $p$ and $q$ are integers to be determined. [6]
\end{enumerate}
The region R is rotated through $2\pi$ radians about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine the volume of the solid of revolution formed. Give your answer in the form $\pi\left(a + b\ln\left(\frac{c}{d}\right)\right)$ where $a$, $b$, $c$ and $d$ are integers to be determined. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q8 [12]}}