2 The diagram shows the curve with equation \(y = \sqrt { ( x - 2 ) ^ { 5 } }\) for \(x \geqslant 2\). \includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-2_885_1125_854_461} The shaded region \(R\) is bounded by the curve \(y = \sqrt { ( x - 2 ) ^ { 5 } }\), the \(x\)-axis and the lines \(x = 3\) and \(x = 4\). Find the exact value of the volume of the solid formed when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

## Question 2:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $V = (\pi)\int y^2\, dx = (\pi)\int(x-2)^5\, dx$ | M1 | |
| $= (\pi)\left[\frac{(x-2)^6}{6}\right]_3^4$ | A1 | Limits not required |
| $= (\pi)\left(\frac{2^6}{6} - \frac{1}{6}\right)$ | m1 | Correct substitution into $(\pi)k(x-2)^6$ |
| $= 10.5\pi$ | A1 | Allow equivalent fraction $\left(\frac{63}{6}\pi\right)$; AWRT 10.5 or $10.5\pi$, m1, A0 |

**Total: 4 marks**

---

2 The diagram shows the curve with equation $y = \sqrt { ( x - 2 ) ^ { 5 } }$ for $x \geqslant 2$.\\
\includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-2_885_1125_854_461}

The shaded region $R$ is bounded by the curve $y = \sqrt { ( x - 2 ) ^ { 5 } }$, the $x$-axis and the lines $x = 3$ and $x = 4$.

Find the exact value of the volume of the solid formed when the region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis.

\hfill \mbox{\textit{AQA C3 2009 Q2 [4]}}