Challenging +1.8 This is a Further Maths question requiring surface area of revolution with parametric equations. Students must apply the formula S = 2π∫y√((dx/dt)²+(dy/dt)²)dt, differentiate trigonometric functions with chain rule, simplify expressions involving sin⁴t and cos⁴t, and integrate the resulting expression. While the formula application is standard for FP1, the algebraic manipulation and integration of the resulting trigonometric expression requires careful work across multiple steps, making it significantly harder than typical A-level questions but not exceptionally difficult for Further Maths.
2 The curve \(C\) is defined parametrically by
$$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$
where \(a\) is a positive constant. Find the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis.
2 The curve $C$ is defined parametrically by
$$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$
where $a$ is a positive constant. Find the area of the surface generated when $C$ is rotated through one complete revolution about the $x$-axis.
\hfill \mbox{\textit{CAIE FP1 2004 Q2 [5]}}