2 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_633_787_402_680} The diagram shows the curve \(y = 3 x ^ { \frac { 1 } { 4 } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\). Find the volume of the solid obtained when this shaded region is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).

## Question 2:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $V = \pi \int 9\sqrt{x}\, dx$ | M1 | For integral of $y^2$ (ignore $\pi$ here) |
| $= \pi \dfrac{9x^{\frac{3}{2}}}{\frac{3}{2}}$ | A1 | All correct (ignoring $\pi$ here) |
| $[\;]$ at $4 - [\;]$ at $1 \rightarrow 42\pi$ | DM1, A1 [4] | Correct use of correct limits; co |

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2\\
\includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_633_787_402_680}

The diagram shows the curve $y = 3 x ^ { \frac { 1 } { 4 } }$. The shaded region is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 4$. Find the volume of the solid obtained when this shaded region is rotated completely about the $x$-axis, giving your answer in terms of $\pi$.

\hfill \mbox{\textit{CAIE P1 2007 Q2 [4]}}