Question 6 - A-Level Maths

CAIE FP1 2012 November — Question 6 7 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2012
Session	November
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Parametric integration
Type	Cartesian area rotated about y-axis
Difficulty	Challenging +1.8 This is a Further Maths surface area of revolution question requiring parametric differentiation, substitution into the formula S = 2π∫x√(1+(dx/dy)²)dy (or equivalent with parameter), and integration of a non-trivial expression. While the setup is standard for FP1, the algebraic manipulation and integration with the logarithmic term make this significantly harder than typical A-level questions, though it follows a well-defined procedure without requiring novel insight.
Spec	4.08d Volumes of revolution: about x and y axes

6 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { 4 } t ^ { 4 } - \ln t$$ for $1 \leqslant t \leqslant 2$. Find the area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $y$-axis.

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Question 6:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$\dot{x} = 2t$, $\dot{y} = t^3 - \frac{1}{t}$	B1	Differentiates
$(\dot{x})^2 + (\dot{y})^2 = 4t^2 + t^6 - 2t^2 + \frac{1}{t^2} = \left(t^3 + \frac{1}{t}\right)^2$	M1A1	Obtains $\left(\frac{ds}{dt}\right)^2$
$S = \int 2\pi x\,ds = 2\pi\int_1^2 t^2\left(t^3+\frac{1}{t}\right)dt = 2\pi\int_1^2(t^5+t)\,dt$	M1A1	Uses surface area formula about $y$-axis
$= 2\pi\left[\frac{1}{6}t^6 + \frac{1}{2}t^2\right]_1^2 = 2\pi\left\{\left[\frac{32}{3}+2\right]-\left[\frac{1}{6}+\frac{1}{2}\right]\right\} = 24\pi$	M1A1

## Question 6:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\dot{x} = 2t$, $\dot{y} = t^3 - \frac{1}{t}$ | B1 | Differentiates |
| $(\dot{x})^2 + (\dot{y})^2 = 4t^2 + t^6 - 2t^2 + \frac{1}{t^2} = \left(t^3 + \frac{1}{t}\right)^2$ | M1A1 | Obtains $\left(\frac{ds}{dt}\right)^2$ |
| $S = \int 2\pi x\,ds = 2\pi\int_1^2 t^2\left(t^3+\frac{1}{t}\right)dt = 2\pi\int_1^2(t^5+t)\,dt$ | M1A1 | Uses surface area formula about $y$-axis |
| $= 2\pi\left[\frac{1}{6}t^6 + \frac{1}{2}t^2\right]_1^2 = 2\pi\left\{\left[\frac{32}{3}+2\right]-\left[\frac{1}{6}+\frac{1}{2}\right]\right\} = 24\pi$ | M1A1 | |

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6 The curve $C$ has parametric equations

$$x = t ^ { 2 } , \quad y = \frac { 1 } { 4 } t ^ { 4 } - \ln t$$

for $1 \leqslant t \leqslant 2$. Find the area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $y$-axis.

\hfill \mbox{\textit{CAIE FP1 2012 Q6 [7]}}

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