| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Integrate using hyperbolic substitution |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question involving surface area of revolution, arc length, and hyperbolic substitution. Parts (a)-(c) are standard applications of formulas requiring careful algebra but no novel insight. Part (d) requires executing a hyperbolic substitution and simplifying, which is a routine FP3 technique. The question is methodical rather than conceptually challenging, placing it moderately above average difficulty. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_6}
The curve $C$ shown in Fig. 1 has equation $y^2 = 4x$, $0 \leq x \leq 1$.
The part of the curve in the first quadrant is rotated through $2\pi$ radians about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Show that the surface area of the solid generated is given by
$$4\pi \int_0^1 \sqrt{1+x} \, dx.$$ [4]
\item Find the exact value of this surface area. [3]
\item Show also that the length of the curve $C$, between the points $(1, -2)$ and $(1, 2)$, is given by
$$2 \int_0^1 \sqrt{\frac{x+1}{x}} \, dx.$$ [3]
\item Use the substitution $x = \sinh^2 \theta$ to show that the exact value of this length is
$$2[\sqrt{2} + \ln(1 + \sqrt{2})].$$ [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q6 [16]}}