Moderate -0.5 This is a straightforward volume of revolution question requiring direct application of the standard formula V = π∫y² dx from x=0 to x=2. The integration of (x³+1) is routine, involving only basic polynomial integration. While it requires careful algebraic manipulation and multiple steps for full marks, it demands no problem-solving insight beyond recognizing the standard technique.
2
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The diagram shows part of the curve \(y = \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }\) and the point \(P ( 2,3 )\) lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Attempt to resolve \(y^2\) and attempt to integrate
\((\pi)\left[\frac{x^4}{4}+x\right]\)
A1
\(6\pi\) or \(18.8\)
DM1A1 [4]
Applying limits 0 and 2. Limits reversed: Allow M mark and allow A mark if final answer is \(6\pi\)
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(\pi)\int(x^3+1)\,dx$ | M1 | Attempt to resolve $y^2$ and attempt to integrate |
| $(\pi)\left[\frac{x^4}{4}+x\right]$ | A1 | |
| $6\pi$ or $18.8$ | DM1A1 [4] | Applying limits 0 and 2. Limits reversed: Allow M mark and allow A mark if final answer is $6\pi$ |
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2\\
\includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-2_627_551_429_790}
The diagram shows part of the curve $y = \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$ and the point $P ( 2,3 )$ lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis.
\hfill \mbox{\textit{CAIE P1 2016 Q2 [4]}}