3 The graph shows part of the curve \(y ^ { 2 } = ( x - 1 )\). \includegraphics[max width=\textwidth, alt={}, center]{73112db3-7b05-48db-9fff-fdbac7dbd564-2_428_860_973_616} Find the volume when the area between this curve, the \(x\)-axis and the line \(x = 5\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\pi\int_1^5 y^2\,dx$ | M1 | |
| $= \pi\int_1^5 (x-1)\,dx$ | A1 | |
| $= \pi\left[\frac{x^2}{2} - x\right]_1^5 = 8\pi$ | B1, A2 | B1 limits; A1 for one part |
| $(= 25.13...)$ | A1 | |
| Volume is $8\pi$ units$^3$ | | |
| **Total: 6** | | |

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3 The graph shows part of the curve $y ^ { 2 } = ( x - 1 )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{73112db3-7b05-48db-9fff-fdbac7dbd564-2_428_860_973_616}

Find the volume when the area between this curve, the $x$-axis and the line $x = 5$ is rotated through $360 ^ { \circ }$ about the $x$-axis.

\hfill \mbox{\textit{OCR MEI C4  Q3 [6]}}