Challenging +1.8 This is a multi-part Further Maths question requiring arc length calculation (with proof), surface area of revolution, and a limit involving centroids. While each technique is standard for FP1, the combination of three distinct concepts, the algebraic manipulation required (especially simplifying the arc length integral and evaluating the limit), and the need to work with the given integral for the centroid calculation make this significantly harder than average A-level questions. However, the methods are all bookwork applications without requiring novel geometric insight.
The curve \(C\) has equation
$$y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } } + \lambda ,$$
where \(\lambda > 0\) and \(0 \leqslant x \leqslant 3\). The length of \(C\) is denoted by \(s\). Prove that \(s = 2 \sqrt { } 3\).
The area of the surface generated when \(C\) is rotated through one revolution about the \(x\)-axis is denoted by \(S\). Find \(S\) in terms of \(\lambda\).
The \(y\)-coordinate of the centroid of the region bounded by \(C\), the axes and the line \(x = 3\) is denoted by h. Given that \(\int _ { 0 } ^ { 3 } y ^ { 2 } \mathrm {~d} x = \frac { 3 } { 4 } + \frac { 8 \sqrt { } 3 } { 5 } \lambda + 3 \lambda ^ { 2 }\), show that
$$\lim _ { \lambda \rightarrow \infty } \frac { S } { h s } = 4 \pi$$
The curve $C$ has equation
$$y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } } + \lambda ,$$
where $\lambda > 0$ and $0 \leqslant x \leqslant 3$. The length of $C$ is denoted by $s$. Prove that $s = 2 \sqrt { } 3$.
The area of the surface generated when $C$ is rotated through one revolution about the $x$-axis is denoted by $S$. Find $S$ in terms of $\lambda$.
The $y$-coordinate of the centroid of the region bounded by $C$, the axes and the line $x = 3$ is denoted by h. Given that $\int _ { 0 } ^ { 3 } y ^ { 2 } \mathrm {~d} x = \frac { 3 } { 4 } + \frac { 8 \sqrt { } 3 } { 5 } \lambda + 3 \lambda ^ { 2 }$, show that
$$\lim _ { \lambda \rightarrow \infty } \frac { S } { h s } = 4 \pi$$
\hfill \mbox{\textit{CAIE FP1 2006 Q12 OR}}