Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x - x ) = \ln x\).
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In the diagram, \(C\) is the curve \(y = \ln x\). The region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = \mathrm { e }\).
(a) Find the exact volume of the solid of revolution formed by rotating \(R\) completely about the \(x\)-axis.
(b) The region \(R\) is rotated completely about the \(y\)-axis. Explain why the volume of the solid of revolution formed is given by
$$\pi \mathrm { e } ^ { 2 } - \pi \int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 y } \mathrm {~d} y ,$$
and find this volume.