Challenging +1.2 This is a volume of revolution problem requiring integration of (2sin x + cosec x)² from π/6 to π/2. While it involves 8 marks and requires expanding the square, using trigonometric identities (double angle formulas), and careful integration of multiple terms, the techniques are all standard C4 content. The 'show that' format provides a target to work towards, reducing problem-solving demand. The algebraic manipulation is moderately involved but follows predictable patterns for this topic.
\includegraphics{figure_1}
Figure 1 shows the curve with equation \(y = 2\sin x + \cosec x\), \(0 < x < \pi\).
The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\) is rotated through \(360°\) about the \(x\)-axis.
Show that the volume of the solid formed is \(\frac{1}{2}\pi(4\pi + 3\sqrt{3})\). [8]
\includegraphics{figure_1}
Figure 1 shows the curve with equation $y = 2\sin x + \cosec x$, $0 < x < \pi$.
The shaded region bounded by the curve, the $x$-axis and the lines $x = \frac{\pi}{6}$ and $x = \frac{\pi}{2}$ is rotated through $360°$ about the $x$-axis.
Show that the volume of the solid formed is $\frac{1}{2}\pi(4\pi + 3\sqrt{3})$. [8]
\hfill \mbox{\textit{Edexcel C4 Q4 [8]}}