Differentiate \(( x - 1 ) ^ { 4 }\) with respect to \(x\).
The diagram shows the curve with equation \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\) for \(x \geqslant 1\).
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The shaded region \(R\) is bounded by the curve \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\), the lines \(x = 2\) and \(x = 4\), and the \(x\)-axis.
Find the exact value of the volume of the solid formed when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Describe a sequence of two geometrical transformations that maps the graph of \(y = \sqrt { x ^ { 3 } }\) onto the graph of \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\).