| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Topic | Volumes of Revolution |
| Type | Rotation about y-axis, region between two curves |
| Difficulty | Challenging +1.2 This is a volumes of revolution question requiring rotation about the y-axis for a region between two curves. Students must express x² in terms of y for both curves, apply the formula V = π∫(x₁² - x₂²)dy, and evaluate the definite integral. While it involves multiple steps and requires careful algebraic manipulation, the setup is straightforward with simple quadratic functions, and the integration itself is routine polynomial integration. This is moderately above average difficulty due to the y-axis rotation and two-curve setup, but remains a standard textbook-style question. |
| Spec | 1.08f Area between two curves: using integration |
Rearrange to obtain $x^2 = 9 - 3y$ and $x^2 = 9 - 5y$ **B1**
Use their $(\pi)\int x^2\,(\text{d}y)$ on separate integrals **M1**
Obtain $9y - \dfrac{3}{2}y^2$ and $9y - \dfrac{5}{2}y^2$ **A1**
Use limits $(3, 0)$ and $(1.8, 0)$ on separate integrals in correct order **M1**
Obtain $13.5\pi$ and $8.1\pi$ **A1**
Subtract separate volumes in correct order **M1**
Obtain $\dfrac{27\pi}{5}$ or equiv ($16.96$ or $5.4\pi$) **A1**
**Alternative method:**
Form a single integral by subtraction $y = \dfrac{1}{3}(9-x^2) - \dfrac{1}{5}(9-x^2) = \dfrac{2}{15}(9-x^2)$ **M1A1**
Rearrange to $x^2$ form $\left(x^2 = 9 - \dfrac{15y}{2}\right)$ **M1**
Use $(\pi)\int x^2\,\text{d}y$ **M1**
Obtain $9y - \dfrac{15}{4}y^2$ **A1**
Use limits $(1.2, 0)$ on a single integral in correct order **M1**
Obtain $\dfrac{27\pi}{5}$ **A1**
**Total: 7 marks**10\\
\includegraphics[max width=\textwidth, alt={}, center]{a3cad2ad-e06b-4aa4-a3a9-a2840cd54893-3_529_527_264_810}
The diagram shows the region $R$ in the first quadrant bounded by the curves $y = \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right)$ and $y = \frac { 1 } { 5 } \left( 9 - x ^ { 2 } \right)$. $R$ is rotated through $360 ^ { \circ }$ about the $y$-axis. Calculate the volume of the solid formed.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2017 Q10 [7]}}