CAIE P1 2020 March — Question 3 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2020
SessionMarch
Marks4
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Mark schemeDownload PDF ↗
TopicVolumes of Revolution
TypeRotation about y-axis, standard curve
DifficultyStandard +0.3 This is a straightforward volume of revolution question about the y-axis requiring students to rearrange y = x² + 1 to get x² = y - 1, then apply the standard formula V = π∫x²dy with clear limits (1 to 5). It involves routine algebraic manipulation and integration of a polynomial, making it slightly easier than average but still requiring proper technique.
Spec4.08d Volumes of revolution: about x and y axes
3 \includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-04_700_401_260_870} The diagram shows part of the curve with equation \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the \(y\)-axis and the line \(y = 5\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Find the volume obtained.
Question 3:
AnswerMarks Guidance
\((\pi)\int(y-1)\,dy\)\*M1 SOI. Attempt to integrate \(x^2\) or \((y-1)\)
\((\pi)\left[\frac{y^2}{2}-y\right]\)A1
\((\pi)\left[\left(\frac{25}{2}-5\right)-\left(\frac{1}{2}-1\right)\right]\)DM1 Apply limits \(1 \to 5\) to an integrated expression
\(8\pi\) or AWRT 25.1A1
Total: 4 
## Question 3:

| $(\pi)\int(y-1)\,dy$ | \*M1 | SOI. Attempt to integrate $x^2$ or $(y-1)$ |
|---|---|---|
| $(\pi)\left[\frac{y^2}{2}-y\right]$ | A1 | |
| $(\pi)\left[\left(\frac{25}{2}-5\right)-\left(\frac{1}{2}-1\right)\right]$ | DM1 | Apply limits $1 \to 5$ to an integrated expression |
| $8\pi$ or AWRT 25.1 | A1 | |
| **Total: 4** | | |

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3\\
\includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-04_700_401_260_870}

The diagram shows part of the curve with equation $y = x ^ { 2 } + 1$. The shaded region enclosed by the curve, the $y$-axis and the line $y = 5$ is rotated through $360 ^ { \circ }$ about the $\boldsymbol { y }$-axis.

Find the volume obtained.\\

\hfill \mbox{\textit{CAIE P1 2020 Q3 [4]}}
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