| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Substitution then integration by parts |
| Difficulty | Challenging +1.8 This FP3 question requires multiple advanced techniques: deriving a surface of revolution formula with parametric equations, performing a substitution to cos θ, then using hyperbolic substitution followed by integration by parts. While each step follows established methods, the multi-stage process and hyperbolic integration make it significantly harder than standard A-level questions, though not exceptionally difficult for Further Maths students who have practiced these techniques. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.08h Integration by substitution4.08d Volumes of revolution: about x and y axes8.06b Arc length and surface area: of revolution, cartesian or parametric |
8. A curve, which is part of an ellipse, has parametric equations
$$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$
The curve is rotated through $2 \pi$ radians about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Show that the area of the surface generated is given by the integral
$$k \pi \int _ { 0 } ^ { a } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \text { where } c = \cos \theta$$
and where $k$ and $\alpha$ are constants to be found.
\item Using the substitution $c = \frac { 3 } { 4 } \sinh u$, or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q8 [8]}}