1. “Teach A Level Maths” Vol. 2: A2 Core Modules
29: Volumes of Revolution
© Christine Crisp

2. Module C3 Module C4 AQA Edexcel OCR MEI/OCR
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3. We’ll first look at the area between the lines
y = x ,
x = 1, . . .
and the x-axis. 1
Can you see what shape you will get if you rotate the area through about the x-axis?
Ans: A cone ( lying on its side )

4. We’ll first look at the area between the lines
y = x ,
x = 1, . . . r
and the x-axis. h 1
For this cone,

5. The formula for the volume found by rotating any area about the x-axis is
where is the curve forming the upper edge of the area being rotated.
a and b are the x-coordinates at the left- and right-hand edges of the area.
We leave the answers in terms of

6. r h So, for our cone, using integration, we get
We must substitute for y using before we integrate. r h 1
I’ll outline the proof of the formula for you.

7. The formula can be proved by splitting the area into narrow strips
. . . x
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 volume

8. The formula can be proved by splitting the area into narrow strips
. . . x
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 volume

9. The formula can be proved by splitting the area into narrow strips
. . . x
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 volume
The formula comes from adding an infinite number of these elements.

10. e.g. 1(a) The area formed by the curve and the x-axis from x = 0 to x = 1 is rotated through radians about the x-axis. Find the volume of the solid formed.
(b) The area formed by the curve , the x-axis and the lines x = 0 and x = 2 is rotated through radians about the x-axis. 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.

11. (a) rotate the area between
rotate about the x-axis
area
A common error in finding a volume is to get wrong. So beware!

12. (a) rotate the area between

14. (b) Rotate the area between
and the lines x = 0 and x = 2.

15. (b) Rotate the area between
and the lines x = 0 and x = 2.

16. Remember that

17. Exercise
radians about the x-axis. Find the volume of the solid formed.
1(a) The area formed by the curve the x-axis and the lines x = 1 to x = 2 is rotated through
(b) The area formed by the curve , the x-axis and the lines x = 0 and x = 2 is rotated through radians about the x-axis. Find the volume of the solid formed.

18. Solutions: 1.
(a)
, the x-axis and the lines x = 1 and x = 2.

19. Solutions:

20. (b)
, the x-axis and the lines x = 0 and x = 2.
Solution:

21. Students taking the EDEXCEL spec do not need to do the next ( final ) section.

22. Rotation about the y-axis
To rotate an area about the y-axis we use the same formula but with x and y swapped.
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.

23. e.g. The area bounded by the curve , the y-axis and the line y = 2 is rotated through about the y-axis. Find the volume of the solid formed.

25. Exercise
the y-axis and the line y = 3 is rotated through
radians about the y-axis. Find the volume of the solid formed.
1(a) The area formed by the curve for
(b) The area formed by the curve , the y-axis and the lines y = 1 and y = 2 is rotated
through radians about the y-axis. Find the volume of the solid formed.

26. Solutions:
(a)
for , the y-axis and the line y = 3.

27. (b)
, the y-axis and the lines y = 1 and y = 2.
Solution:

29. 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.

30. x
The formula for the volume found by rotating any area about the x-axis is
a and b are the x-coordinates at the left- and right-hand edges of the area.
where is the curve forming the upper edge of the area being rotated. a b
We leave the answers in terms of

31. e.g. 1 Find the volume of the solid formed by rotating through about the x-axis the area bounded by the given curves and lines.
(b)
, 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.
(a)
and the x-axis from x = 0 to x = 1.

32. a = 0, b = 1 (a) rotate the area between rotate about the x-axis area
A common error in finding a volume is to get wrong. So beware!
(a) rotate the area between
a = 0, b = 1

34. (b) Rotate the area between
and the lines x = 0 and x = 2.

35. Remember that

36. STUDENTS TAKING THE EDEXCEL SPEC DO NOT NEED THIS SECTION.
To rotate an area about the y-axis we use the same formula but with x and y swapped.
The limits of integration are now values of y giving the top and bottom of the area that is rotated.
Rotation about the y-axis
As we have to substitute for x from the equation of the curve we will have to rearrange the equation.

37. e.g. The area bounded by the curve , the y-axis and the line y = 2 is rotated through about the y-axis. Find the volume of the solid formed.