12 In this question you must show detailed reasoning.
- Given that \(y = \arctan x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
Fig. 12 shows the curve \(y = \frac { 1 } { 1 + x ^ { 2 } }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09d39832-5519-463d-ac7a-5d406ffd7be0-5_444_1435_1371_332}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{figure} - Find, in exact form, the mean value of the function \(\mathrm { f } ( x ) = \frac { 1 } { 1 + x ^ { 2 } }\) for \(- 1 \leq x \leq 1\).
- The region bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = - 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated.