Moderate -0.3 This is a straightforward C3 volume of revolution question with standard setup. Part (a) is trivial differentiation, part (b) requires the standard formula V=π∫y²dx with simple algebraic manipulation (y²=4(x-1)³), and part (c) tests basic transformations. The integration is routine with no substitution needed, making this slightly easier than the average A-level question.
Differentiate \(( x - 1 ) ^ { 4 }\) with respect to \(x\).
The diagram shows the curve with equation \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\) for \(x \geqslant 1\).
\includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-02_789_1180_1190_431}
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 \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\).
2
\begin{enumerate}[label=(\alph*)]
\item Differentiate $( x - 1 ) ^ { 4 }$ with respect to $x$.
\item The diagram shows the curve with equation $y = 2 \sqrt { ( x - 1 ) ^ { 3 } }$ for $x \geqslant 1$.\\
\includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-02_789_1180_1190_431}
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.
\item Describe a sequence of two geometrical transformations that maps the graph of $y = \sqrt { x ^ { 3 } }$ onto the graph of $y = 2 \sqrt { ( x - 1 ) ^ { 3 } }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2007 Q2 [9]}}