CHAPTER 74 VOLUMES OF SOLIDS OF REVOLUTION€¦ · CHAPTER 74 VOLUMES OF SOLIDS OF REVOLUTION . EXERCISE 288 Page 783 . 1. Determine the volume of the solid of revolution formed by
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1. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curve, the x-axis and the given ordinates through one revolution about the x-axis: y = 5x ; x = 1, x = 4 A sketch of y = 5x is shown below.
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
given by: volume = ( )4 4 422 2
1 1 1d 5 d 25 dy x x x x xπ π π= =∫ ∫ ∫
= 43
1
64 1 6325 25 253 3 3 3xπ π π = − =
= 525π cubic units
2. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curve, the x-axis and the given ordinates through one revolution about the x-axis: y = x2 ; x = –2, x = 3 A sketch of y = 2x is shown below.
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
given by: volume = ( )3 3 322 2 42 2 2
d d dy x x x x xπ π π− − −
= =∫ ∫ ∫
= 35
2
243 32 2755 5 5 5xπ π π
−
= − − = = 55π cubic units
3. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curve, the x-axis and the given ordinates through one revolution about the x-axis:
y = 2x3 + 3 ; x = 0, x = 2
A sketch of y = 32 3x + is shown below
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
given by: volume = ( ) ( )2 2 222 2 4 2
0 0 0d 2 3 d 4 12 9 dy x x x x x xπ π π= + = + +∫ ∫ ∫
= ( )25 3
0
4 12 1289 32 18 05 3 5x x xπ π + + = + + −
= ( )25.6 32 18π + + = 75.6π cubic units
4. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curve, the x-axis and the given ordinates through one revolution about the x-axis:
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
given by: volume = 525 5
21 1
1
25 1d 4 d 4 42 2 2xy x x xπ π π π = = = −
∫ ∫
= 48π cubic units
5. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curve, the x-axis and the given ordinates through one revolution about the x-axis: xy = 3 ; x = 2, x = 3
A sketch of xy = 3, i.e. 3yx
= is shown below.
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
6. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curves, the y-axis and the given ordinates through one revolution about the y-axis: y = x2 ; y = 1, y = 3
A sketch of y = x2 is shown below
When the shaded area is rotated one revolution about the y-axis
volume = 3
21
dx yπ∫
Since y = x2 , then x = y
Hence, volume = ( )323 3
21 1
1
( ) d d (4.5) 0.52yy y y xπ π π π = = = − ∫ ∫ = 4π cubic units
7. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curves, the y-axis and the given ordinates through one revolution about the y-axis: y = 3x2 – 1; y = 2, y = 4 A sketch of y = 23 1x − , is shown below
Revolving the shaded area shown one revolution about the y-axis produces a solid of revolution
given by: volume = ( ) ( )424 4
22 2
2
1d d 8 4 2 23 3 2 3
y yx y y yπ ππ π + = = + = + − + ∫ ∫
= [ ]83π = 2.67π cubic units
8. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curves, the y-axis and the given ordinates through one revolution about the y-axis:
y = 2x
; y = 1, y = 3
A sketch of y = 2x
, is shown below.
Revolving the shaded area shown one revolution about the y-axis produces a solid of revolution
given by: volume = ( )2 313 3 3
2 21 1 1
1
2 1d d 4 d 4 4 11 3
yx y x y yy
π π π π π−
− = = = = − − −
∫ ∫ ∫
= 243
π − − = 2.67π cubic units
9. The curve y = 2x2 + 3 is rotated about (a) the x-axis between the limits x = 0 and x = 3, and (b) the y-axis, between the same limits. Determine the volume generated in each case. (a) The curve is shown below.
1. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curve, the x-axis and the given ordinates through one revolution about the x-axis: y = 4ex ; x = 0, x = 2
A graph of y = 4ex lies wholly above the x-axis. Revolving y = 4ex one revolution about the x-axis produces a solid of revolution given by:
volume = ( ) [ ]222 2 222 2 4 0
0 0 00
ed 4e d 16 e d 16 8 e e2
xx xy x x xπ π π π π = = = = − ∫ ∫ ∫
= 428.8π cubic units
2. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curve, the x-axis and the given ordinates through one revolution about the x-axis:
y = sec x ; x = 0, x = 4π
A sketch of part of y = sec x is shown below
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
given by: volume = [ ]/4 /4 /42 2
00 0d sec d tan tan tan 0
4y x x x x
π π π ππ π π π = = = − ∫ ∫
= [ ]1 0π − = π cubic units
3. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curves, the y-axis and the given ordinates through one revolution about the y-axis: x2 + y2 = 16 ; y = 0, y = 4
A sketch of x2 + y2 = 16 , i.e. x2 + y2 = 42 is shown below
Revolving the shaded area shown one revolution about the y-axis produces a solid of revolution
given by: volume = ( ) ( )434 4
2 20 0
0
64d 16 d 16 64 03 3yx y y y yπ π π π = − = − = − −
∫ ∫
= 42.67π cubic units
4. Determine the volume of the solid of revolution formed by revolving the area enclosed by the curves, the y-axis and the given ordinates through one revolution about the y-axis: x y = 2 ; y = 2, y = 3
A sketch x y = 2, i.e. 2yx
= and y = 2
4x
is shown below.
Revolving the shaded area shown one revolution about the y-axis produces a solid of revolution
given by: volume = [ ] [ ]3 3 32
22 2
4 3d d 4 ln 4 ln 3 ln 2 4 ln2
x y y yy
π π π π π= = = − =∫ ∫
= 1.622π cubic units
5. Determine the volume of a plug formed by the frustum of a sphere of radius 6 cm which lies between two parallel planes at 2 cm and 4 cm from the centre and on the same side of it (the equation of a circle, centre 0, radius r is x2 + y2 = r2).
6. The area enclosed between the two curves x2 = 3y and y2 = 3x is rotated about the x-axis. Determine the volume of the solid formed. The curves are shown in the sketch below
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
given by: volume = ( )34 2 53 3
20 0
0
3 27 243d 3 d 09 2 45 2 45x x xy x x xπ π π π = − = − = − −
lying between x = 1 and x = 3 is revolved 360° about the x-
axis. Determine the volume of the solid formed.
A sketch of part of 2 1y xx
= + , is shown below
Revolving the shaded area shown one revolution about the x-axis produces a solid of revolution
given by: volume = 2 35 13 3 3
2 2 4 221 1 1
1
1 1d d 2 d5 1x xy x x x x x x x
x xπ π π π
− = + = + + = + + − ∫ ∫ ∫
= 243 1 19 1 15 3 5
π + − − + − = 57.07π cubic units
8. Calculate the volume of the frustum of a sphere of radius 5 cm that lies between two parallel planes at 3 cm and 2 cm from the centre and on opposite sides of it. The volume of a frustum of a sphere may be determined by integration by rotating the curve
x2 + y2 = 52 (i.e. a circle, centre 0, radius 5) one revolution about the x-axis, between the limits x = 3
and x = –2 (i.e. rotating the shaded area of sketch below)
9. Sketch the curves y = x2 + 2 and y – 12 = 3x from x = –3 to x = 6. Determine (a) the coordinates of the points of intersection of the two curves, and (b) the area enclosed by the two curves. (c) If the enclosed area is rotated 360° about the x-axis, calculate the volume of the solid produced. The curves are shown sketched below
(a) Equating the y-values gives: 2 2x + = 3x + 12
i.e. 2 3 10 0x x− − =
and (x – 5)(x + 2) = 0
from which, x = 5 and x = –2
When x = –2, y = 6 and when x = 5, y = 27
Hence, the points of intersection of the two curves are at (–2, 6) and (5, 27)