Challenging +1.8 This is an FP3 surface area of revolution question with parametric equations requiring the formula S = 2π∫y√((dx/dt)² + (dy/dt)²)dt. Students must differentiate trigonometric functions with chain rule, simplify expressions involving sin⁴t + cos⁴t using identities, and integrate products of trigonometric functions. While the parametric setup and algebraic manipulation demand sophistication beyond standard C3/C4, the question follows a predictable template for FP3 with 8 marks indicating extended working but no exceptional conceptual leaps.
\includegraphics{figure_31}
Figure 1 shows a sketch of the curve with parametric equations
$$x = a \cos^3 t, \quad y = a \sin^3 t, \quad 0 \leq t \leq \frac{\pi}{2},$$
where \(a\) is a positive constant.
The curve is rotated through \(2\pi\) radians about the \(x\)-axis. Find the exact value of the area of the curved surface generated. [8]
\includegraphics{figure_31}
Figure 1 shows a sketch of the curve with parametric equations
$$x = a \cos^3 t, \quad y = a \sin^3 t, \quad 0 \leq t \leq \frac{\pi}{2},$$
where $a$ is a positive constant.
The curve is rotated through $2\pi$ radians about the $x$-axis. Find the exact value of the area of the curved surface generated. [8]
\hfill \mbox{\textit{Edexcel FP3 Q31 [8]}}