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8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
1/20
4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
Integration4U97-1a)!
Evaluate2
0
5
x + 4dx .
4 4U97-1b)!
Evaluatesin
cosd
4
0
4
.
13
2 2 1
4U97-1c)!
Find1
2x x + 3dx
2 .
1
2
tan x +1
2
C-1
4U97-1d)!
Find4t -6
(t +1)(2tdt
2 3).
Lnc (2t
t +1)
2
2
3)
(
4U97-1e)!
Evaluate x sec x dx2
0
3
.
3
3 Ln2
4U96-1a)!
Evaluate4
(2 x)dx
21
3
.
8
15
4U96-1b)!
Find sec tan2
d .
1
2tan2+ c
4U96-1c)!
Find5t 3
t(t 1)dt
2
2
.
3 Ln t+ Ln (t2+ 1) + c 4U96-1d)!
Using integration by parts, or otherwise, find x x dxtan 1 .
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
2/20
4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
1
2(x2tan-1x + tan-1xx) + c
4U96-1e)!
Using the substitution x 2sin , or otherwise, calculatex
4 xdx
2
21
3
.
3
4U96-3b)!
i. Show that (sin ) cosx xdxk
k2
0
2 1
2 1
, where k is a positive integer.
ii. By writing (cos ) ( sin )x xn n2 21 , show that (cos ) ( )
x dxk
n
kn
k
k
n2 1
00
21
2 1
.
iii. Hence, or otherwise, evaluate cos5
0
2
xdx
.
i) Proof ii) Proof iii)8
15
4U96-7b)!
i. Let f(x) = Lnx - ax + b, for x > 0, where a and b are real numbers and a > 0. Show that
y = f(x) has a single turning point which is a maximum.
ii. The graphs of y = Lnx and y = ax - b intersect at points A and B. Using the result of part (i),
or otherwise, show that the chord AB lies below the curve y = Lnx.
iii. Using integration by parts, or otherwise, show that1
k
Lnx dx = k Lnk - k + 1.
iv. Use the trapezoidal rule on the intervals with integer endpoints 1, 2, 3,..., k to show that
Lnx dx
k
1
is approximately equal to 1
2 1Lnk Ln k ( )! .
v. Hence deduce that k e k k
e
k
!
.
Proof 4U95-1a)!
Find dx
x(lnx)2
.
1
ln xc
4U95-1b)!
Find xedxx .ex- xex+ c
4U95-1c)!
Show that6t 23
(2t 1)(t 6)dt ln 70
1
4
.
Proof
4U95-1d)!
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
3/20
4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
Findd
dx(x sin x)1 , and hence find sin x dx1 .
x
1 xsin x
2
1
, x sin x 1 x c1 2
4U95-1e)!
Using the substitution t = tanx
2, or otherwise, calculate
dx
5 3sin x 4cos x02
.
1
6
4U95-4c)!
i. Show that, if 0 < x Ik + 1.
iv. Hence, using the fact that I2n - 1> I2nand I2n> I2n + 1, show that
2
2n2n 1
2 .4 ...(2n)
1.3 .5 ...(2n 1) (2n 1)
2 2 2
2 2 2
2.
Proof 4U94-1a)!
Find
i.4 12
6 132x
x x dx
ii.1
6 132x x dx
.
(i) 2log ce x x2 6 13 (ii) 12 1tan x 3
2c
4U94-1b)!
Evaluate 9 232
3
u du .
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
i. Show that f(x) f 2 x 1 for 0 x 2
.
ii. Sketch y = f(x) for 0 x2
.
[There is no need to find f xbut assume y = f(x) has a horizontal tangent at x = 0. Yourgraph should exhibit the property of (b)(i).]
iii. Hence, or otherwise, evaluate
dx
1 (tan x)k02
.
(i) Proof (ii)2
1
y
x (iii)
4
4U92-1a)!
Find:
i. tan sec 2 d .
ii.2 6
6 12x
x x dx
.(i) 12
2tan c (ii) logx 6x 1 c2
4U92-1b)!
Evaluatedx
(x 1)(3 x)32
5
2
by using the substitution u = x - 2.
3
4U92-1c)!
Evaluate5 1
1 3 20
1 ( )
( )( )
t
t t dt .
log 4
3e
4U92-1d)!
i. Find xe dxx2
.
ii. Evaluate 2x e dx3 x0
1 2
.(i) 12
xe c2
(ii) 1 4U92-4a)!
Each of the following statements is either true or false. Write TRUE or FALSE for each statement and
give brief reasons for your answers. (You are not asked to find the primitive functions).
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
i. sin7
2
2
0
d
ii. sin70
0
d
iii. e x
2
1
10dx
iv. (sin80
2 0
- cos ) d8
v. For n = 1, 2, 3, ...., dt
tdttn n1 10
1
10
1
.(i) True (ii) False (iii) False (iv) True (v) True
4U91-1a)!
Find:
i.t1
t3
dt ;
ii.e
x
e2x 9 dx , using the substitution u = ex.
(i) 1
t
1
2t c2 (ii)
13tan
e
3 c1
x
4U91-1b)!
i. Evaluate5
(2t 1)(2t)dt
0
1
.
ii. By using the substitution t = tan
2 and (i), evaluate
d
3 40
2
sin cos
.
(i) ln 6 (ii)1
5ln 6
4U91-1c)!
Let I sin x dxnn
0
2
, where n is a non-negative integer.
i. Show that I (n 1) sin xcos x dxnn 2 2
0
2
when n 2.
ii. Deduce that In n 1
nIn2 when n 2.
iii. Evaluate I4.
(i) Proof (ii) Proof (iii) 316
4U90-2a)!
Find the exact values of:
i.x 1
x 2x 5dx
22
3
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
7/20
4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
ii. 4 x dx2
0
2
(i) 2 5 13 (ii)
21
4U90-2b)!
Find:
i.dx
(x 1)(x2 2) ;ii. cos x dx3 , by writing cos
3x(1sin2 x)cosx , or otherwise.
(i) 13 162 1ln x 1 ln(x 2)
1
3 2tan
x
2c (ii) sin x sin x c13
3
4U90-2c)!
Let I (1 t ) dt, n 1,2,3, .. .n2
0
xn
Use integration by parts to show that I 1
2n 1
(1 x ) x 2n
2n 1
I 1n2 n
n
.
Hint: Observe that (1 t2)n1 t2(1 t2)n 1 (1 t2)n .Proof
4U89-2a)!
Evaluate:
i.2
920
3
x dx ;
ii. x lnx dx2
1
3
;
iii. 4 x dx2
3
2
.
(i) 9
(ii) 9 3 26
9ln (iii)
33
2
4U89-2b)!
i. Write4x2 5x 7
(x 1)(x2 x 2)in the form
A
x 1
Bx C
x2 x 2
.
ii. Hence evaluate4x2 5x 7
(x1)(x2 x 2)dx
1 .
(i) 2
x 16x 3
x x 22 (ii) 2 ln 2
4U88-1a)!
Find the exact value of:
i.2x
1 2xdx
0
1
;
ii.(logex)
2
xdx
1
e
;
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
8/20
4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
iii. cos x dx1
0
1
2
.
(i) 1 312 loge (ii)1
3 (iii)
1
3
2
4U88-1b)!
Use the substitution t tan2
to find the exact value ofd
sin 20
2
.
3 3
4U87-1i)Prove that:
a.t 1
t2 dt
1
2
1
2loge2;
b.
4 dx
(x 1)(x 3) 2log
9
5e4
6
.Proof
4U87-1ii)
a. Use the substitution x2
3sin to prove that 4 9x dx
32
0
2
3
.
b. Hence, or otherwise, find the area enclosed by the ellipse 9x2+ y2= 4.
(a) Proof (b) 43
units2
4U87-1iii)
a. Given that I x dxnn
cos0
2
, prove that I n 1
nIn n 2
where n is and integer and n 2.
b. Hence evaluate cos x dx5
0
2
.
(a) Proof (b)8
15
4U86-1i)
Evaluatex
xdx
( )20
1
.
23
4 2 - 5
4U86-1ii)
Use integration by parts to show that tan x dx4
12
log 21
0
1
e .
Proof 4U86-1iii)
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
Find numbers A, B, C such thatx
4x A
B
2x 3
C
2x 3
2
2 9
.
Hence evaluatex
2
4x2 9dx
0
.
A 1
4 , B
3
8 , C = -
3
8;
1
4
3
16log 5e
4U86-1iv)
Using the substitution t = tan(1
2 ), or otherwise, show that
1
1 3 1
0
3
sin d .
Proof 4U85-1i)
Find:
a. x cos x dx0
3
;
b. 1x2 4x 5
dx0
.
(a) -2 (b) 0.1419 (to 4 decimal places) 4U85-1ii)
Find real numbers A, B, C such thatx
(x 1)2(x 2)
A
x 1
B
(x 1)2
C
x 2.
Hence show thatx
(x 1) (x 2)dx 2log
3
21
2
0
1
2
e
.
A = -2, B = -1, C = 2, Proof 4U85-1iii)!
Use the substitution x = a - t, where a is a constant, to prove that f(x)dx f(a t)dt0
a
0
a
. Hence, or
otherwise, show that x x dx( )1 1
10 10099
0
1
.
Proof 4U84-1i)
Show that:
a. 4 xlog x dx 7e ee8 2
e
e4
;
b.dx
xloge x2 loge
e
e4
2 .
(a) Proof (b) Proof 4U84-1ii)
Evaluate:
a.x 1
x 1dx
2
0
3
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSINTEGRATION HSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
b.x 6
x 5dx
4
4
c. sin 4x cos 2x dx
6
6
.
(a) log 2 3e
(b) 2113 (c) 0
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
Volumes4U97-3a)!
y
xO
(3, 2)C(0, 2)
In the diagram, the shaded region is bounded by the x axis, the line x = 3, and the circle with centre
C(0, 2) and radius 3. Find the volume of the solid formed when the region is rotated about the y
axis.
8
3units3
4U97-5a)!
i. Sketch the graph of y = -x2for 2 x 2.
ii. Hence, without using calculus, sketch the graph of y = e-x2
for 2 x 2.
iii. The region between the curve y = e-x2
, the x axis, the y axis, and the line x = 2 is rotated
around the y axis to form a solid. Using the method of cylindrical shells, find the volume of
the solid.
i)
Y
-4
-2 2X
ii)
1
y
xO-2 2
iii) 1 1
eunits
4
3
4U96-3a)!
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
x + y = 0
0
-1
1
41 x
S
The shaded region is bounded by the lines x = 1, y = 1, and y = -1 and by the curve x y 2 0 . The
region is rotated through 360about the line x = 4 to form a solid. When the region is rotated, the line
segment S at height y sweeps out an annulus.
i. Show that the area of the annulus at height y is equal to ( )y y4 28 7 .
ii. Hence find the volume of the solid. i) Proof ii)
296
15 units3
4U95-3b)!
The circle x + y is rotated about the line x = 9 to form a ring.
When the circle is rotated, the line segment S at height y sweeps out an annulus.
0 4
s
4
-4
-4 x
y
x = 9
The x coordinates of the end-points are x1and - x1, where x1= 16 2 y Error! Switch argument
not specified..
i. Show that the area of the annulus is equal to 36 16 2 y Error! Switch argument not
specified.
ii. Hence find the volume of the ring.
(i) Proof (ii) 2882units3
4U94-3b)!
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
-2 x
y
2 4 6
0
E
t = 0
The graph shows part of the curve whose parametric equations are
x t t 2 2 , y t t 3 , t0 .. i. Find the values of t corresponding to the points O and E on the curve.
ii. The volume V of the solid formed by rotating the shaded area about the y axis is to be
calculated using cylindrical shells.
Express V in the form V f ta
b 2 ( ) dt .
Specify the limits of integration a and b and the function f(t). You may leave f(t) in
unexpanded form.
Do NOT evaluate this integral.
(i) t = 1, 2 (ii) V 2 (t t 2)(t t)(2t 1) dt2 31
2
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
4U93-6a)!
The solid S is generated by moving a straight edge so that it is always parallel to a fixed plane P. It is
constrained to pass through a circle C and a line segment L. The circle C, which forms a base for S,
has radius a and the line segment L is distance d from C. Both C and L are perpendicular to P and sit
on P in such a way that the perpendicular C at its centre O bisects L.
P
O
lS
F
G
E
C
a
d
x
i. Calculate the area of the triangular cross-section EFG which is parallel to P and distance x
from the centre O of C.
ii. Calculate the volume of S.
(i) a x d2 2 units2 (ii) 12 a2d units3
4U92-5a)!
2a
axa
a The solid S is a rectangular prism of dimensions a a 2a from which right square pyramids of basea a and height a have been removed from each end. The solid T is a wedge that has been obtained
by slicing a right circular cylinder of radius a at 45through a diameter AB of its base.Consider a cross-section of S which is parallel to its square base at distance x from its centre, and a
corresponding cross-section of T which is perpendicular to AB and at distance x from its centre.
i. The triangular cross-section of T is shown on the diagram.
Show that its area is
1
2
2 2
( )a x
.
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
ii. Draw the cross-section of S and calculate its area.
iii. Express the volumes of S and T as definite integrals.
iv. What is the relationship between the volumes of S and T? (There is no need to evaluate
either integral).
(i) Proof (ii)
x
a
A B
C
A = a2- x2 (iii) V 2 (a x ) dxS2 2
0
a
,
V (a x ) dxT2 2
0
a
. (iv) V VT 12 S
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
4U91-5b)!
E
B
O
Figure 1
Water surface
D
C
A F
G
H
H
E
a
C'
FG
Water
Figure 2
Figure 3
ha
F D
O
C
H
A drinking glass having the form of a right circular cylinder of radius a and height h, is filled withwater. The glass is slowly tilted over, spilling water out of it, until it reaches the position where the
waters surface bisects the base of the glass. Figure 1 shows this position.
In Figure 1, AB is a diameter of the circular base with centre C, O is the lowest point on the base, and
D is the point where the waters surface touches the rim of the glass.
Figure 2 shows a cross-section of the tilted glass parallel to its base. The centre of this circular section
is Cand EFG shows the water level. The section cuts the lines CD and OD of Figure 1 in F and H
respectively.
Figure 3 shows the section COD of the tilted glass.
i. Use Figure 3 to show that FH =a
h(h x) , where OH = x.
ii. Use Figure 2 to show that C'F = axh
and HC'G = cos-1 x
h .
iii. Use (ii) to show that the area of the shaded segment EGH is
a cos x
h
x
h1
x
h
2 12
.
iv. Given that cos d cos 11 1 2 , find the volume of water in the tilted glass ofFigure 1.
(i) Proof (ii) Proof (iii) Proof (iv)2a h
3
2
units3
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
4U90-3a)!
x
y
y = e - x
- a a
S
t
The region under the curve yex
2
and above the x axis for a x a is rotated about the y axis toform a solid.
i. Divide the resulting solid into cylindrical shells S of radius t as in the diagram. Show that
each shell S has approximate volume V 2tet2
t where t is the thickness of the shell.
ii. Hence calculate the volume of the solid.iii. What is the limiting value of the volume of the solid as a approaches infinity?
(i) Proof (ii) (1 e )a2
units3 (iii) units34U89-5b)!
P
OZ
XY
B
T
a
A Let ABO be and isosceles triangle, AO = BO = r, AB = b.
Let PABO be a triangular pyramid with height OP = h and OP perpendicular to the plane of ABO as in
the diagram. Consider a slice S of the pyramid of width a as in the diagram. The slice s is
perpendicular to the plane of ABO at XY with XY || AB and XB = a. Note that XT || OP.
i. Show that the volume of S isr a
r b
ahr
a
, when a is small. (You may assume that the
slice is approximately a rectangular prism of base XYZT and height a.)
ii. Hence show that the pyramid PABO has volume1
6hbr.
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
iii. Suppose now that AOB =2
nand that n identical pyramids PABO are arranged about O as
centre with common vertical axis OP to form a solid C. Show that the volume Vnof C is
given by Vn1
3r
2hnsin
n.
iv. Note that when n is large, the solid C approximates a right circular cone. Using the fact thatsinx
x
approaches 1 as x approaches 0, find limn
Vn. Hence verify that a right circular cone of
radius r and height h has volume1
3r
2h .
(i) Proof (ii) Proof (iii) Proof (iv) 13r h2
4U88-7a)!
K
L
H
J
M
I20cm
xcm
16cm
4cm
hcm
FIGURE NOT TO SCALE
i. A trapezium HIJK has parallel sides KJ = 16 and HI = 20cm. The distance between these
sides is 4cm. L lies on HK and M lies on IJ such that LM is parallel to KJ. The shortest
distance from K to LM is h cm and LM has length x cm. Prove that x = 16 + h.
ii.
20cm
16cm
12cm
10cm
FIGURE NOT TO SCALE
The diagram above is of a cake tin with a rectangular base with sides of 16cm and 10cm. Its
top is also rectangular with dimensions 20cm and 12cm. The tin has depth 4cm and each of
its four side faces is a trapezium. Find its volume.
(i) Proof (ii) 794 cm233
4U87-5i)
a. On a number plane shade the region representing the inequality (x - 2R)2+ y2R2.
b. Show that the volume of a right cylindrical shell of height h with inner and outer radii x and
x + x respectively is 2xhx when x is sufficiently small for terms involving (x)2to be
neglected.
c. The region (x - 2R)2+ y2R2is rotated about the y axis forming a solid of revolution called
a torus. By summing volumes of cylindrical shells show that the volume of the torus is given
by: V = 42R3.
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
BOARD OF STUDIESNSW1984-1997
EDUDATA:DATAVER1.01995
(a)
2R
R 3R x
y
(x - 2R)2+ y2R2 (b) Proof (c) Proof
4U86-5i)
a. The coordinates of the vertices of a triangle ABC are (0, 2), (1, 1), (-1, 1) respectively and His the point (0, h). The line through H, parallel to the x-axis meets AB and AC at X and Y.
Find the length of XY.
b. The region enclosed by the triangle ABC, defined in (a) above, is rotated about the x axis.
Find the volume of the solid of revolution so formed.
(a) (4 - 2h) units (b)8
3
units3
4U85-5ii)
a. Using the substitution x = asin, or otherwise, verify that (a2 x2)
1
2 dx1
4a2
0
a
.
b. Deduce that the area enclosed by the ellipsex
a
y
b1
2
2
2
2 is ab.
c.
h
H
The diagram shows a mound of height H. At height h above the horizontal base, the
horizontal cross-section of the mound is elliptical in shape, with equationx
a
y
b
2
2
2
2
2 ,
where 1h
2
H2, and x, y are appropriate coordinates in the plane of the cross-section.
Show that the volume of the mound is8abH
15.
Proof
4U84-5)
8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4
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4UNIT MATHEMATICSVOLUMESHSC
x
y
- a 0 a
t t
S
i. The diagram shows the area A between the smooth curve y = f(x), -a x a, and the x axis.
(Note that f(x) 0 for -a x a and f(-a) = f(a) = 0). The area A is rotated about the line
x = -s (where s > a) to generate the volume V. This volume is to be found by slicing A into
thin vertical strips, rotating these to obtain cylindrical shells, and adding the shells. Two
typical strips of width t whose centre lines are distance t from the y axis are shown.
a. Show that the indicated strips generate shells of approximate volume 2f(-t)(s -
t)t, 2f(t)(s + t)t, respectively.
b. Assuming that the graph of f is symmetrical about the y axis, show that V = 2sA.ii. Assuming the results of part (i), solve the following problems.
a. A doughnut shape is formed by rotating a circular disc of radius r about and axis in
its own plane at a distance of s(s > r) from the centre of the disc. Find the volume
of the doughnut.
b. The shape of a certain party jelly can be represented by rotating the area between
the curve y = sinx, 0 x , and the x axis about the line x =
4. Find the
volume generated.
(i)(a) Proof (b) Proof (ii)(a) 22r2s units3 (b) 3
2units3
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