1. Volumes of Revolution

2. Volumes of Revolution
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 )

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

4. Volumes of Revolution
The formula for the volume found by rotating any area about the x-axis is x a b
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

5. r h So, for our cone, using integration, we get
Volumes of Revolution
So, for our cone, using integration, we get
We must substitute for y using before we integrate. r h 1
Now we’ll outline the proof of the formula ...

6. The formula can be proved by splitting the area into narrow strips
Volumes of Revolution
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

7. The formula can be proved by splitting the area into narrow strips
Volumes of Revolution
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
Volumes of Revolution
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.

9. through radians about the x-axis. Find the volume of the solid formed.
Volumes of Revolution
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 we’ll sketch the curves.

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

11. Volumes of Revolution
(a) rotate the area between
a = 0, b = 1

12. Volumes of Revolution

13. (b) Rotate the area between and the lines x = 0 and x = 2.
Volumes of Revolution
(b) Rotate the area between
and the lines x = 0 and x = 2.

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

15. Volumes of Revolution
Remember that

16. radians about the x-axis. Find the volume of the solid formed.
Volumes of Revolution
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.

17. , the x-axis and the lines x = 1 and x = 2.
Volumes of Revolution
Solutions: 1.
(a)
, the x-axis and the lines x = 1 and x = 2.

18. Volumes of Revolution
Solutions:

19. , the x-axis and the lines x = 0 and x = 2.
Volumes of Revolution
(b)
, the x-axis and the lines x = 0 and x = 2.
Solution:

20. Rotation about the y-axis
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.
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.

21. Volumes of Revolution
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.

22. Volumes of Revolution

23. the y-axis and the line y = 3 is rotated through
Volumes of Revolution
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.

24. For , the y-axis and the line y = 3.
Volumes of Revolution
Solutions:
(a)
For , the y-axis and the line y = 3.

25. , the y-axis and the lines y = 1 and y = 2.
Volumes of Revolution
(b)
, the y-axis and the lines y = 1 and y = 2.
Solution: