5 In this question, \(a\) and \(k\) are positive constants.
The region enclosed by the curve \(y = a \mathrm { e } ^ { - \frac { x } { a } }\) for \(0 \leqslant x \leqslant k a\), the \(x\)-axis, the \(y\)-axis and the line \(x = k a\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of mass \(m\). Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 1 } { 4 } m a ^ { 2 } \left( 1 + \mathrm { e } ^ { - 2 k } \right)\).