Challenging +1.8 This is a demanding Further Maths question requiring surface area of revolution with parametric equations (involving a non-trivial arc length element) and centroid calculation. Both parts require setting up and evaluating challenging integrals with substitution, going well beyond standard A-level techniques.
9 The curve \(C\) has parametric equations
$$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$
Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\).
9 The curve $C$ has parametric equations
$$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$
Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.
Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$.
\hfill \mbox{\textit{CAIE FP1 2013 Q9}}