8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-28_680_1266_118_482}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
- Find \(\int x \cos 4 x d x\)
Figure 3 shows part of the curve with equation \(y = \sqrt { x } \sin 2 x , \quad x \geqslant 0\)
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 4 }\)
The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution. - Find the exact value of the volume of this solid of revolution, giving your answer in its simplest form.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\includegraphics[max width=\textwidth, alt={}, center]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-32_2630_1828_121_121}