3 At any point \(( x , y )\) on the curve \(C\), $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t \sqrt { } \left( t ^ { 2 } + 4 \right) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = - t \sqrt { } \left( 4 - t ^ { 2 } \right)$$ where the parameter \(t\) is such that \(0 \leqslant t \leqslant 2\). Show that the length of \(C\) is \(4 \sqrt { } 2\). Given that \(y = 0\) when \(t = 2\), determine the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis, leaving your answer in an exact form.

$\frac{ds}{dt} = \sqrt{t(t^2 + 4) + t^2(4-t)} = \sqrt{8t^2} = 2\sqrt{2}t$ | B1 |

$s = 2\sqrt{2} \int_0^t tdt = 2\sqrt{2}[t^2]_0 = 4\sqrt{2}$ (AG) | M1A1 |

| [3] |

$y = (1/3)(4-t^2)^{3/2}$ | B1 |

$S = +2\pi/3 \int_0^2 2\sqrt{2t(4-t^2)^{3/2}}dt$ | M1 |

$= ... = [-(4\sqrt{2\pi}/15)(4-t^2)^{5/2}]_0$ | A1 |

$= 128\sqrt{2\pi}/15$ | A1 |

| [4] |

3 At any point $( x , y )$ on the curve $C$,

$$\frac { \mathrm { d } x } { \mathrm {~d} t } = t \sqrt { } \left( t ^ { 2 } + 4 \right) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = - t \sqrt { } \left( 4 - t ^ { 2 } \right)$$

where the parameter $t$ is such that $0 \leqslant t \leqslant 2$. Show that the length of $C$ is $4 \sqrt { } 2$.

Given that $y = 0$ when $t = 2$, determine the area of the surface generated when $C$ is rotated through one complete revolution about the $x$-axis, leaving your answer in an exact form.

\hfill \mbox{\textit{CAIE FP1 2010 Q3 [7]}}