Show that the derivative with respect to \(y\) of
$$y \ln ( 2 y ) - y$$
is \(\ln ( 2 y )\).
\includegraphics[max width=\textwidth, alt={}, center]{390105da-0cba-4f82-8c8f-1f36090b1564-3_465_631_1859_717}
The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \mathrm { e } ^ { x ^ { 2 } }\). The point \(P \left( 2 , \frac { 1 } { 2 } \mathrm { e } ^ { 4 } \right)\) lies on the curve. The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = \frac { 1 } { 2 } e ^ { 4 }\). Find the exact volume of the solid produced when the shaded region is rotated completely about the \(y\)-axis.
Hence find the volume of the solid produced when the region bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\) is rotated completely about the \(y\)-axis.
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