OCR M4 2004 June — Question 3 6 marks

Exam BoardOCR
ModuleM4 (Mechanics 4)
Year2004
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCentre of Mass 2
TypeCentre of mass of solid of revolution
DifficultyChallenging +1.2 This is a standard M4/Further Mechanics centre of mass question requiring the formula x̄ = ∫x·y²dx / ∫y²dx with straightforward polynomial integration after expanding (3-x)x². While it involves multiple steps and careful algebra, it follows a well-practiced template with no conceptual surprises, making it moderately above average difficulty for A-level but routine for Further Maths students.
Spec1.08h Integration by substitution6.04d Integration: for centre of mass of laminas/solids
3 The region between the curve \(y = x \sqrt { } ( 3 - x )\) and the \(x\)-axis for \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.
Question 3:
AnswerMarks Guidance
\(\int \pi xy^2\,dx = \int_0^3 \pi x^3(3-x)\,dx\)M1 (\(\pi\) may be omitted throughout)
\(= \pi\left[\frac{3}{4}x^4 - \frac{1}{5}x^5\right]_0^3 \quad (= 12.15\pi)\)A1
\(\int \pi y^2\,dx = \int_0^3 \pi x^2(3-x)\,dx\)M1
\(= \pi\left[x^3 - \frac{1}{4}x^4\right]_0^3 \quad (= 6.75\pi)\)A1
\(\bar{x} = \frac{12.15\pi}{6.75\pi}\)M1 Dependent on previous M1M1
\(= 1.8\)A1 Total: 6 
# Question 3:

$\int \pi xy^2\,dx = \int_0^3 \pi x^3(3-x)\,dx$ | M1 | ($\pi$ may be omitted throughout)
$= \pi\left[\frac{3}{4}x^4 - \frac{1}{5}x^5\right]_0^3 \quad (= 12.15\pi)$ | A1 |
$\int \pi y^2\,dx = \int_0^3 \pi x^2(3-x)\,dx$ | M1 |
$= \pi\left[x^3 - \frac{1}{4}x^4\right]_0^3 \quad (= 6.75\pi)$ | A1 |
$\bar{x} = \frac{12.15\pi}{6.75\pi}$ | M1 | Dependent on previous M1M1
$= 1.8$ | A1 | **Total: 6**

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3 The region between the curve $y = x \sqrt { } ( 3 - x )$ and the $x$-axis for $0 \leqslant x \leqslant 3$ is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid of revolution. Find the $x$-coordinate of the centre of mass of this solid.

\hfill \mbox{\textit{OCR M4 2004 Q3 [6]}}
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Q1 4 Q2 6 Q3 6 Q4 9 Q5 10 Q6 11 Q7 14