Standard +0.3 Part (a) is a standard integration by parts with exponential function (routine C3/C4 content). Part (b) requires recognizing that squaring the function gives xe^(6x) from part (a), then applying the volume of revolution formula—a straightforward connection between parts with minimal additional work. The question guides students through the solution structure, making it slightly easier than average.
By using integration by parts, find \(\int x \mathrm { e } ^ { 6 x } \mathrm {~d} x\).
(4 marks)
The diagram shows part of the curve with equation \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\).
\includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-4_547_846_536_591}
The shaded region \(R\) is bounded by the curve \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\), the line \(x = 1\) and the \(x\)-axis from \(x = 0\) to \(x = 1\).
Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in the form \(\pi \left( p \mathrm { e } ^ { 6 } + q \right)\), where \(p\) and \(q\) are rational numbers.
(3 marks)
4
\begin{enumerate}[label=(\alph*)]
\item By using integration by parts, find $\int x \mathrm { e } ^ { 6 x } \mathrm {~d} x$.\\
(4 marks)
\item The diagram shows part of the curve with equation $y = \sqrt { x } \mathrm { e } ^ { 3 x }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-4_547_846_536_591}
The shaded region $R$ is bounded by the curve $y = \sqrt { x } \mathrm { e } ^ { 3 x }$, the line $x = 1$ and the $x$-axis from $x = 0$ to $x = 1$.
Find the volume of the solid generated when the region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis, giving your answer in the form $\pi \left( p \mathrm { e } ^ { 6 } + q \right)$, where $p$ and $q$ are rational numbers.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2012 Q4 [7]}}