Standard +0.8 This is a two-part surface area of revolution question requiring arc length formula derivation followed by algebraic manipulation to express surface area in terms of the arc length. While the techniques are standard for Further Maths, the algebraic manipulation in the second part (factoring out constants and simplifying the expression involving s) requires careful work and is more demanding than typical C3/C4 questions. The 'show that' format and multi-step nature place it above average difficulty.
4 A curve has equation
$$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$
The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\). Show that
$$s = \int _ { 0 } ^ { 1 } \sqrt { } \left( 1 + x ^ { 4 } \right) \mathrm { d } x$$
The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\). Show that
$$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { } 2 - 1 )$$
[Do not attempt to evaluate \(s\) or \(S\).]
4 A curve has equation
$$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$
The length of the arc of the curve joining the point where $x = 0$ to the point where $x = 1$ is denoted by $s$. Show that
$$s = \int _ { 0 } ^ { 1 } \sqrt { } \left( 1 + x ^ { 4 } \right) \mathrm { d } x$$
The surface area generated when this arc is rotated through one complete revolution about the $x$-axis is denoted by $S$. Show that
$$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { } 2 - 1 )$$
[Do not attempt to evaluate $s$ or $S$.]
\hfill \mbox{\textit{CAIE FP1 2009 Q4 [6]}}