| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Volume of revolution with parts |
| Difficulty | Standard +0.3 This is a standard C3 integration question testing routine techniques: integration by parts (textbook example with x ln x), chain rule differentiation of composite logarithm, and volume of revolution requiring substitution of y² = x ln x. All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.08d Evaluate definite integrals: between limits1.08i Integration by parts |
\begin{enumerate}[label=(\alph*)]
\item Use integration by parts to find $\int x\ln x \, dx$. [3]
\item Given that $y = (\ln x)^2$, find $\frac{dy}{dx}$. [2]
\item The diagram shows part of the curve with equation $y = \sqrt{x\ln x}$.
\includegraphics{figure_9}
The shaded region $R$ is bounded by the curve $y = \sqrt{x\ln x}$, the line $x = e$ and the $x$-axis from $x = 1$ to $x = e$.
Find the volume of the solid generated when the region $R$ is rotated through 360° about the $x$-axis, giving your answer in an exact form. [6]
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2011 Q9 [11]}}