12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-40_695_1212_276_420}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve with equation
$$y = x ( \sin x + \cos x ) , \quad 0 \leqslant x \leqslant \frac { \pi } { 4 }$$
The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\). This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution, with volume \(V\).
- Assuming the formula for volume of revolution show that \(V = \int _ { 0 } ^ { \frac { \pi } { 4 } } \pi x ^ { 2 } ( 1 + \sin 2 x ) \mathrm { d } x\)
- Hence using calculus find the exact value of \(V\).
You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)