| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Surface area of revolution: Cartesian curve |
| Difficulty | Standard +0.8 This FP3 question requires deriving the surface area formula for a sphere using calculus (surface of revolution), which is conceptually more sophisticated than standard C3/C4 volumes of revolution. Part (a) involves implicit differentiation and algebraic manipulation, part (b) requires setting up and evaluating a non-trivial integral for surface area (8 marks total), and part (c) connects to arc length. While the techniques are within FP3 scope, the multi-step proof and conceptual depth place it moderately above average difficulty. |
| Spec | 1.07s Parametric and implicit differentiation4.08d Volumes of revolution: about x and y axes |
A circle $C$ with centre $O$ and radius $r$ has cartesian equation $x^2 + y^2 = r^2$ where $r$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $1 + \left(\frac{dy}{dx}\right)^2 = \frac{r^2}{r^2 - x^2}$ [3]
\item Show that the surface area of the sphere generated by rotating $C$ through $\pi$ radians about the $x$-axis is $4\pi r^2$. [5]
\item Write down the length of the arc of the curve $y = \sqrt{1 - x^2}$ from $x = 0$ to $x = 1$ [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2014 Q7 [9]}}