A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 3\), and the line \(x = \ln 3\). Find, in an exact form, the coordinates of the centre of mass of this lamina.
A region is bounded by the \(x\)-axis, the curve \(y = \frac { 6 } { x ^ { 2 } }\) for \(2 \leqslant x \leqslant a\) (where \(a > 2\) ), the line \(x = 2\) and the line \(x = a\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.
Show that the \(x\)-coordinate of the centre of mass of this solid is \(\frac { 3 \left( a ^ { 3 } - 4 a \right) } { a ^ { 3 } - 8 }\).
Show that, however large the value of \(a\), the centre of mass of this solid is less than 3 units from the origin.