Standard +0.8 This is a solid of revolution problem requiring integration of sec²(x/2). While the setup is standard, students must correctly apply the volume formula πy² and handle the sec function integration (requiring substitution u=x/2 and knowledge that ∫sec²u du = tan u), then evaluate at the limits. The trigonometric integration and exact form requirement elevate this above routine calculus questions but it remains a recognizable Further Maths technique.
In this question you must show detailed reasoning.
Fig. 4 shows the region bounded by the curve \(y = \sec \frac{1}{2}x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{2}\pi\).
\includegraphics{figure_4}
This region is rotated through \(2\pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.
[3]
In this question you must show detailed reasoning.
Fig. 4 shows the region bounded by the curve $y = \sec \frac{1}{2}x$, the $x$-axis, the $y$-axis and the line $x = \frac{1}{2}\pi$.
\includegraphics{figure_4}
This region is rotated through $2\pi$ radians about the $x$-axis.
Find, in exact form, the volume of the solid of revolution generated.
[3]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q3 [3]}}