Challenging +1.2 This is a standard Further Maths mechanics question requiring integration to find the centre of mass of a composite region. While it involves multiple steps (finding area, first moment, and dealing with two curves), the techniques are routine for FM students: setting up integrals with respect to y (since bounded by y-axis), finding intersection points (given), and applying standard formulae. The algebra is manageable and no novel geometric insight is required.
4 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242}
The diagram shows parts of the curves \(y = 3 \sqrt { x }\) and \(y = 4 - x ^ { 2 }\), which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the \(y\)-axis, is occupied by a uniform lamina.
Determine the exact \(x\)-coordinate of the centre of mass of the lamina.
A = ∫ ( 4 − x ) − 3 x dx = 4x − x3 − 2x 2 3 1
1 2 1
0 3
0
1 5
A = 4 − − 2 =
3 3
1
1 3 1 6 5
3 2 4
Ax = ∫ 4x − x − 3x 2 dx = 2x − x − x 2
0 4 5
0
1 6 11
Ax = 2 − − =
4 5 20
Ax 11
x = = 20
A 5
3
33
=
Answer
Marks
100
M1*
A1
M1*
A1
M1dep*
A1
Answer
Marks
[6]
2.1
1.1
1.1
1.1
1.1
Answer
Marks
2.2a
Correct integral expression for the area
and attempt to integrate (at least two
terms correct)
Correct integral expression for Ax and
attempt to integrate (at least two terms
correct)
Ax
Correct use of x =
A
Answer
Marks
oe
Ignore limits for first
two M marks
SC M1 A0 if correct
integral and value
seen but with no
intermediate working
SC M1 A0 if correct
integral and value
seen but with no
intermediate working
Dependent on both
previous M marks
This mark can be
awarded even if the
two previous A marks
were not awarded
Question 4:
4 | DR
A = ∫ ( 4 − x ) − 3 x dx = 4x − x3 − 2x 2 3 1
1 2 1
0 3
0
1 5
A = 4 − − 2 =
3 3
1
1 3 1 6 5
3 2 4
Ax = ∫ 4x − x − 3x 2 dx = 2x − x − x 2
0 4 5
0
1 6 11
Ax = 2 − − =
4 5 20
Ax 11
x = = 20
A 5
3
33
=
100 | M1*
A1
M1*
A1
M1dep*
A1
[6] | 2.1
1.1
1.1
1.1
1.1
2.2a | Correct integral expression for the area
and attempt to integrate (at least two
terms correct)
Correct integral expression for Ax and
attempt to integrate (at least two terms
correct)
Ax
Correct use of x =
A
oe | Ignore limits for first
two M marks
SC M1 A0 if correct
integral and value
seen but with no
intermediate working
SC M1 A0 if correct
integral and value
seen but with no
intermediate working
Dependent on both
previous M marks
This mark can be
awarded even if the
two previous A marks
were not awarded
4 In this question you must show detailed reasoning.\\
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242}
The diagram shows parts of the curves $y = 3 \sqrt { x }$ and $y = 4 - x ^ { 2 }$, which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the $y$-axis, is occupied by a uniform lamina.
Determine the exact $x$-coordinate of the centre of mass of the lamina.
\hfill \mbox{\textit{OCR MEI Further Mechanics Major 2021 Q4 [6]}}