| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Centre of mass of lamina by integration |
| Difficulty | Challenging +1.2 This is a standard Further Maths centroid and surface of revolution question requiring multiple integrations with exponential functions. Part (i) involves routine centroid formulas with straightforward exponential integrals. Part (ii) requires showing a given arc length derivative (which simplifies nicely due to the hyperbolic cosine identity) and computing a Pappus theorem or direct surface area integral. While it requires several steps and careful algebra with exponentials, the techniques are all standard for FM students and the expressions simplify cleanly throughout. Slightly above average difficulty due to the multi-part nature and FM content, but no novel insight required. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08d Volumes of revolution: about x and y axes6.04d Integration: for centre of mass of laminas/solids8.06b Arc length and surface area: of revolution, cartesian or parametric |
The curve $C$ has equation $y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)$ for $0 \leqslant x \leqslant 4$.\\
(i) The region $R$ is bounded by $C$, the $x$-axis, the $y$-axis and the line $x = 4$. Find, in terms of e, the coordinates of the centroid of the region $R$.\\
(ii) Show that $\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)$, where $s$ denotes the arc length of $C$, and find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q12 EITHER}}