| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Stationary points then area/volume |
| Difficulty | Standard +0.3 This is a standard C3 integration by parts question with routine differentiation for stationary points. Part (a) requires product rule and solving a quadratic (7 marks suggests straightforward execution). Part (b)(i) is textbook double integration by parts, and (b)(ii) is a direct application using the given answer. All techniques are standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.08i Integration by parts4.08d Volumes of revolution: about x and y axes |
7
\begin{enumerate}[label=(\alph*)]
\item A curve has equation $y = x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } }$.\\
Show that the curve has exactly two stationary points and find the exact values of their coordinates.\\
(7 marks)
\item \begin{enumerate}[label=(\roman*)]
\item Use integration by parts twice to find the exact value of $\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } } \mathrm {~d} x$.
\item The region bounded by the curve $y = 3 x \mathrm { e } ^ { - \frac { x } { 8 } }$, the $x$-axis from 0 to 4 and the line $x = 4$ is rotated through $360 ^ { \circ }$ about the $x$-axis to form a solid.
Use your answer to part (b)(i) to find the exact value of the volume of the solid generated.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2012 Q7 [16]}}