Mathematics · Oct / Nov 2025 · Paper 41 · Question 2
YMA01
As printed in the paper
The question as text
y 2. C R x O 2 8 Figure 1 Figure 1 shows a sketch of the curve C with equation 5 y x 1 3 The region R, shown shaded in Figure 1, is bounded by C, the line with equation x = 2, the x‑axis and the line with equation x = 8 This region is then rotated through 2π radians about the x-axis to form a solid. Use algebraic integration to find the exact volume of this solid. (5)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
+
5
x
4
4
16
three
term quadratic but ignore any terms in
3x or higher.
May just state e.g.
=
= −
=
a
"4",
b
" 20",
c
"75"
They must all be values (not unevaluated expressions)
25 (
π
x
−
1) d
25
π
(
)
(
)
Volume =
oe e.g.
−
2
x
x
1 d
x
9
9
2
2
25(
x
−
1)
25(
x
−
1)
25
x
25
x
π
−
( )
π
d
x
= π
( )
or
( )
18
9
9
18
Volume =
8
2
2
2
π
−
π
−
π
−
25 (
x
1)
25 (8 1)
25 (2 1)
200
=
−
=
π
18
18
18
3
2
B1: Correct expression for the volume ignoring limits
25
π
25
π
x
25
π
(
)
(
)
May see e.g.
x
−
1 d
x
or
−
d
x
May be unsimplified so
9
9
9
2
5
(
)
allow
π
x
−
1
d
x
Condone omission of dx. It cannot be scored if they
3
make errors before the integral expression is written. π may appear later in a
correct integral or integrand. Alternatively may use substitution e.g.
u
50
3
(
)
proceeding to
π
oe
9 u
d
u
2
M1: Correct attempt at integrating (
x −
1)
. Look for
A x −
(
1)
2
or
Bx
−
Cx
where A, B
and C are non zero constants (indices do not need to be processed for this
mark). Do not award this mark if this is only part of a sum of integrals. If using the
substitution
2
=
− then look for
3
→
4
u
x
1
u
u
2
2
x −
25
x
25
x
25(
1)
π
−
A1:
( )
π
or
( )
oe (or using the substitution
2
=
u
x
18
9
18
25
( ) 18 u
π
4
oe)
Ignore any reference to π which could even appear as 2π
M1:Uses the limits of 2 and 8 within their integrated expression of the form
8
2
π
−
25 (
x
1)
200
( )
2
−
and subtracts either way round. Allow
= ±
...
x
...
x
18
3
2
score this mark but if their integration is incorrect then we must see evidence of
using limits not just a final answer. May be partially evaluated which you may
need to check. Ignore the absence of π
If using the substitution
=
− then they need to use the limits
7 and 1 on an
u
2
x
1
expression of the form …
u
4
A1: 200
2
3 π or exact equivalent e.g.
66 3π. If they subtract the wrong way round to
200
achieve
−
π they cannot ignore the negative sign but must subsequently
3
state e.g. volume = 200
3 π. Ignore units.
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