“Teach A Level Maths”
Vol. 2: A2 Core Modules
29: Volumes of Revolution
© Christine Crisp
Volumes of Revolution
Module C3
Module C4
AQA
Edexcel
OCR
MEI/OCR
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Volumes of Revolution
We’ll first look at the area between the lines
y=x, ...
x = 1, . . .
and the x-axis.
yx
0
1
Can you see what shape you will get if you rotate the
area through 360  about the x-axis?
Ans: A cone ( lying on its side )
Volumes of Revolution
We’ll first look at the area between the lines
y=x, ...
x = 1, . . .
and the x-axis.
yx
r
0
h
1
V  13  r 2 h
For this cone,
r  1, h  1

V  13 
Volumes of Revolution
The formula for the volume found by rotating any
area about the x-axis is
y  f ( x)
V 

b
y 2 dx
a
x
a
b
where y  f ( x ) is the curve forming the upper edge
of the area being rotated.
a and b are the x-coordinates at the left- and righthand edges of the area.
We leave the answers in terms of

Volumes of Revolution
V 

b
y 2 dx
a
So, for our cone, using integration, we get
We must substitute
for y using y  f ( x ) before
1 2
we integrate.
V   x dx
0
x
 
 3
 1
 
 3
 1
3
3
yx
1


0

 0

r
0
h
I’ll outline the proof of the formula for you.
1
Volumes of Revolution
The formula can be proved by
splitting the area into narrow strips
. . . which are rotated about the
x-axis.
Each tiny piece is approximately a
cylinder ( think of a penny on its
side ).
Each piece, or element, has a
2
volume
2
 r h   y
y
x
Volumes of Revolution
The formula can be proved by
splitting the area into narrow strips
. . . which are rotated about the
x-axis.
Each tiny piece is approximately a
cylinder ( think of a penny on its
side ).
Each piece, or element, has a
2
volume
2
  r h   y dx
y
x
dx
Volumes of Revolution
The formula can be proved by
splitting the area into narrow strips
. . . which are rotated about the
x-axis.
Each tiny piece is approximately a
cylinder ( think of a penny on its
side ).
Each piece, or element, has a
2
volume
2
y
x
dx
  r h   y dx
The formula comes from adding an infinite number
of these elements.
V 

b
a
y 2 dx
Volumes of Revolution
e.g. 1(a) The area formed by the curve y  x(1  x )
and the x-axis from x = 0 to x = 1 is
rotated through 2 radians about the xaxis. Find the volume of the solid formed.
(b) The area formed by the curve y  e x , the
x-axis and the lines x = 0 and x = 2 is
rotated through 2 radians about the xaxis. Find the volume of the solid formed.
Solution: To find a volume we don’t need a sketch
unless we are not sure what limits of integration
we need. However, a sketch is often helpful.
As these are the first examples I’ll sketch the curves.
Volumes of Revolution
(a) rotate the area between
y  x(1  x ) and the x - axis from 0 to 1.
y  x(1  x )
area
V 

b
2
y dx
a
A common error in finding
a volume is to get y 2
wrong. So beware!
rotate about
the x-axis
y  x(1  x )
2
2
2
 y  x (1  x )
y 2  x 2 (1  2 x  x 2 )
2
2
3
4
y  x  2x  x
Volumes of Revolution
(a) rotate the area between
y  x(1  x ) and the x - axis from 0 to 1.
y  x(1  x )
V 

b
y2  x2  2x3  x4
2
y dx
a
1

a = 0, b = 1

 V    x 2  2 x 3  x 4 dx
0
Volumes of Revolution
1


 V    x 2  2 x 3  x 4 dx
0
1
x
2x
x 
 



42 5  0
 3
 1 1 1

        0 
 3 2 5

1

30


30
3
4
5
Volumes of Revolution
(b) Rotate the area between
y  e x and the lines x = 0 and x = 2.
y  ex
x2
V 

b
a
y 2 dx
 
y  e
2
e 
x 2
x 2

 e x  e x  e2x
Volumes of Revolution
(b) Rotate the area between
y  e x and the lines x = 0 and x = 2.
y  ex
x2
V 

b
a
2
y dx
 
y  e
2
2
 V    e 2 x dx
0
x 2
 e2x
Volumes of Revolution
2
V   e
2x
0
dx
2
e 
 

 2 0
2x
0 
 e4
e

  


2
2


Remember that
4
 e4


1
e 1

  
    

2
2
2




e0  1
Volumes of Revolution
Exercise
1
1(a) The area formed by the curve y 
the x-axis
x
and the lines x = 1 to x = 2 is rotated through 2
radians about the x-axis. Find the volume of
the solid formed.
(b) The area formed by the curve y  x x ,
the x-axis and the lines x = 0 and x = 2 is
rotated through 2 radians about the xaxis. Find the volume of the solid formed.
Volumes of Revolution
Solutions:
1 , the x-axis and the lines x = 1 and x = 2.
1. (a) y 
x
V 

b
a
y dx  V  
2

 V  
2
1
2
1
1
x
2
dx
x 2 dx
2
x 
 V  

 1  1
2
 1
 V   
 x 1
1
Volumes of Revolution
Solutions:
2
 1
V   
 x 1



V   

V 

2

 1
  2     1 



Volumes of Revolution
(b)
y  x x , the x-axis and the lines x = 0
and x = 2.
Solution:
V 
yx x

b
2
y dx
y2  x x  x x  x3
a
 V  
2
x
0
3
2
x 
 V   
 4 0
 V  4
4
dx
Volumes of Revolution
Students taking the EDEXCEL spec do not need to
do the next ( final ) section.
Volumes of Revolution
Rotation about the y-axis
To rotate an area about the y-axis we use the
same formula but with x and y swapped.
b
V    y dx
a
2
d
V    x 2 dy
c
Tip: dx for rotating about the x-axis;
dy for rotating about the y-axis.
The limits of integration are now values of y giving
the top and bottom of the area that is rotated.
As we have to substitute for x from the equation of
the curve we will have to rearrange the equation.
Volumes of Revolution
e.g. The area bounded by the curve y  x , the
y-axis and the line y = 2 is rotated through 360 
about the y-axis. Find the volume of the
solid formed.
y2
y
V  
d
c
y x
 x  y2
 x2  y4
x
2
4

V


y
x dy
0 dy
2
Volumes of Revolution
2
V    y 4 dy
0
2
 y 
 

 5 0
 5 

 2

  
   0 
 5 




5

32
5
Volumes of Revolution
Exercise
1(a) The area formed by the curve y  x 2 for x  0
the y-axis and the line y = 3 is rotated through
2 radians about the y-axis. Find the
volume of the solid formed.
1
(b) The area formed by the curve y 
, the
x
y-axis and the lines y = 1 and y = 2 is rotated
through 2 radians about the y-axis. Find the
volume of the solid formed.
Volumes of Revolution
Solutions:
(a)
y  x 2 for x  0 , the y-axis and the line y = 3.
yx
d
V    x dy
2
c
3
 V    y dy
0
3
 y 
9
 
 
2
 2 0
2
2
Volumes of Revolution
1
(b) y  , the y-axis and the lines y = 1 and y = 2.
x
Solution:
d
V    x 2 dy
c
 V 
2
1
1
2
dy
y
2
 1
  
 y 1

  

1
1
y
 x
x
y
1
2
 x  2
y
  1    1   



 2 
2

Volumes of Revolution
Volumes of Revolution
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Volumes of Revolution
The formula for the volume found by rotating any
area about the x-axis is
y  f ( x)
V 

b
y 2 dx
a
x
a
b
where y  f ( x ) is the curve forming the upper edge
of the area being rotated.
a and b are the x-coordinates at the left- and righthand edges of the area.
We leave the answers in terms of

Volumes of Revolution
e.g. 1 Find the volume of the solid formed by
rotating through 360  about the x-axis the
area bounded by the given curves and lines.
(a) y  x(1  x ) and the x-axis from x = 0
to x = 1.
(b) y  e x , the x-axis, and the lines x = 0
and x = 2.
Solution: To find a volume we don’t need a sketch
unless we aren’t sure what limits of integration we
need. However, a sketch is often helpful.
Volumes of Revolution
(a) rotate the area between
y  x(1  x ) and the x - axis from 0 to 1.
y  x(1  x )
area
rotate about
the x-axis
a = 0, b = 1
V 

b
a
y 2 dx
A common error in finding a volume
is to get y 2 wrong. So beware!
y  x(1  x )
 y 2  x 2 (1  x ) 2
2
2
2
y  x (1  2 x  x )
y2  x2  2x3  x4
Volumes of Revolution
1
 V    x 2  2 x 3  x 4 dx
0
1
x
2x
x 
 



42 5  0
 3
 1 1 1

        0 
 3 2 5

1

30


30
3
4
5
Volumes of Revolution
(b) Rotate the area between
y  e x and the lines x = 0 and x = 2.
y  ex
x2
V 

b
a
 
y  e
x 2
y dx  V  
2
2
2
 e2x
0 e
2x
dx
Volumes of Revolution
2
V   e
2x
0
dx
2
e 
 

 2 0
2x
0 
 e4
e

  


2
2


Remember that
4
 e4


1
e 1

  
    

2
2
2




e0  1
Volumes of Revolution
STUDENTS TAKING THE EDEXCEL SPEC DO NOT
NEED THIS SECTION.
Rotation about the y-axis
To rotate an area about the y-axis we use the
same formula but with x and y swapped.
b
V    y dx
a
2
d
V    x 2 dy
c
The limits of integration are now values of y giving
the top and bottom of the area that is rotated.
As we have to substitute for x from the equation of
the curve we will have to rearrange the equation.
Volumes of Revolution
e.g. The area bounded by the curve y  x , the
y-axis and the line y = 2 is rotated through 360 
about the y-axis. Find the volume of the
solid formed.
y2
y
V  
d
c
y x
 x  y2
 x2  y4
x
2
4

V


y
x dy
0 dy
2
Volumes of Revolution
2
V    y 4 dy
0
2
 y 
 

 5 0
 5 

 2

  
   0 
 5 




5

32
5