Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } \sqrt { 8 x ^ { 3 } + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
Describe the single transformation that maps the graph of \(y = \sqrt { 8 x ^ { 3 } + 1 }\) onto the graph of \(y = \sqrt { x ^ { 3 } + 1 }\).
The curve with equation \(y = \sqrt { x ^ { 3 } + 1 }\) is translated by \(\left[ \begin{array} { c } 2 - 0.7 \end{array} \right]\) to give the curve with equation \(y = \mathrm { g } ( x )\). Find the value of \(\mathrm { g } ( 4 )\).
(3 marks)