Challenging +1.8 This is a demanding Further Maths mechanics question requiring: (1) finding the volume of revolution about the y-axis using integration, (2) finding the first moment about the origin, (3) applying the centre of mass formula. Students must invert y = (2/3)ln x to get x = e^(3y/2), set up integrals with correct limits (0 to ln 2), and handle exponential integrals carefully. The 8-mark allocation and 'show detailed reasoning' requirement indicate extended multi-step work with potential for algebraic errors, placing it well above average difficulty.
In this question you must show detailed reasoning.
\includegraphics{figure_7}
Fig. 7 shows the curve with equation \(y = \frac{2}{3}\ln x\). The region R, shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis.
Find the exact \(x\)-coordinate of the centre of mass of the solid. [8]
Question 7:
7 | DR
2
V ln2 e 3 2 y dy
0
ln2
1
e3y
3
0
1 7
e3ln2e0
3 3
2
ln2 3y
Vy ye2 dy
0
ln2
1 ye3y 1 ln2 e3ydy
3 0 3 0
ln2
1 1
ye3y e3y
3 9
0
8 7
ln2
3 9
y
7
3
8 1
y ln2
7 3 | M1*
A1
A1
M1*
A2
M1dep*
A1
[8] | 1.1
1.1
1.1
2.1
1.1
1.1
1.1
2.2a | 2 2
For e 3 2 y dy or e3 2y dy
1e3y
For
3
For correct substitution of limits and
removing of logs
For yx2dy leading to
ye3y e3ydyor consistent with
2
2y
their V if using e3 dy
Both terms integrated correctly (A1
for one error)
Vy
M1 for y - dependent on both
V
previous M marks
oe | Limits not required for
M and first A mark
Must see one line of
working from
integrated expression to
answer
Clear indication of
integrating exponential
term and differentiating
𝑦 term
Limits not required for
M mark and both A
marks
Must have used correct
limits correctly
Allow absence of
throughout
\textbf{In this question you must show detailed reasoning.}
\includegraphics{figure_7}
Fig. 7 shows the curve with equation $y = \frac{2}{3}\ln x$. The region R, shown shaded in Fig. 7, is bounded by the curve and the lines $x = 0$, $y = 0$ and $y = \ln 2$. A uniform solid of revolution is formed by rotating the region R completely about the $y$-axis.
Find the exact $x$-coordinate of the centre of mass of the solid. [8]
\hfill \mbox{\textit{OCR MEI Further Mechanics Major 2019 Q7 [8]}}