| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Surface area of revolution: parametric curve |
| Difficulty | Challenging +1.8 This FP3 question requires parametric arc length and surface of revolution formulas with non-trivial trigonometric integration. Part (a) involves computing arc length using $\int \sqrt{(dx/d\theta)^2 + (dy/d\theta)^2} d\theta$ which simplifies nicely using $\sin^2\theta + \cos^2\theta = 1$, but requires careful manipulation. Part (b) applies the surface area formula $2\pi \int y ds$ which involves similar techniques. While the integrals simplify elegantly with trigonometric identities, this is advanced Further Maths content requiring mastery of parametric calculus and extended multi-step reasoning beyond standard A-level. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_25}
Figure 1 shows the curve with parametric equations
$$x = a \cos^3 \theta, \quad y = a \sin^3 \theta, \quad 0 \leq \theta < 2\pi.$$
\begin{enumerate}[label=(\alph*)]
\item Find the total length of this curve. [7]
\end{enumerate}
The curve is rotated through $\pi$ radians about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the area of the surface generated. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q25 [12]}}