| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Derivative then integrate by parts |
| Difficulty | Challenging +1.2 This is a structured multi-part question on arc length and surface of revolution requiring standard techniques (differentiation of sec x, integration by parts, and using a given derivative result). While it involves several steps and further maths content, the question provides significant scaffolding including the arc length formula to prove and the key derivative identity, making it more accessible than typical FP1 questions requiring independent insight. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)4.08c Improper integrals: infinite limits or discontinuous integrands8.06b Arc length and surface area: of revolution, cartesian or parametric |
The curve $C$ has equation $y = 2 \sec x$, for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$. Show that the arc length $s$ of $C$ is given by
$$S = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 2 } x - 1 \right) d x$$
Find the exact value of $s$.
The surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis is denoted by $S$. Show that\\
(i) $S = 4 \pi \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 3 } x - \sec x \right) \mathrm { d } x$,\\
(ii) $\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x \tan x ) = 2 \sec ^ { 3 } x - \sec x$.
Hence find the exact value of $S$.
\hfill \mbox{\textit{CAIE FP1 2013 Q11 EITHER}}