1. Revolution about x-axis
Solid of Revolution
Revolution about x-axis

2. What is a Solid of Revolution - 1
Consider the area under the graph of y = 0.5x from x = 0 to x = 1:

3. What is a Solid of Revolution - 2
If the shaded area is now rotated about the x-axis, then a three-dimensional solid (called Solid of Revolution) will be formed:
What will it look like?
Pictures from

4. What is a Solid of Revolution - 3
Actually, if the shaded triangle is regarded as made up of straight lines perpendicular to the x-axis, then each of them will give a circular plate when rotated about the x-axis.  The collection of all such plates then pile up to form the solid of revolution, which is a cone in this case.

5. Finding Volume

6. How is it calculated - 1 What will it look like?
Consider the solid of revolution formed by the graph of y = x2 from x = 0 to x = 2:
What will it look like?
victor-victrola-gramophone-II.jpg

7. How is it calculated - 2
Just like the area under a continuous curve can be approximated by a series of narrow rectangles, the volume of a solid of revolution can be approximated by a series of thin circular discs:
we could improve our accuracy by using a larger and larger number of circular discs,
making them thinner and thinner

8. How is it calculated - 3 x x x As n tends to infinity,
It means the discs get thinner and thinner.
And it becomes a better and better approximation.
As n tends to infinity,
It means the discs get thinner and thinner.
And it becomes a better and better approximation.
It can be replaced by an integral

9. Volume of Revolution Formula
The volume of revolution about the x-axis
between x = a and x = b, as , is :
This formula you do need to know
Think of is as the um of lots of circles
… where area of circle = r2

10. Example of a disk r= the y value of the function
How could we find the volume of the cone?
One way would be
to cut it into a series of disks
(flat circular cylinders)
and add their volumes.
The volume of each disk is:
In this case:
r= the y value of the function
thickness = a small change in x = dx

11. The volume of each flat cylinder (disk) is:
If we add the volumes, we get:

12. Integrating and substituting gives:
Example 1
Consider the area under the graph of y = 0.5x from x = 0 to x = 1:
What is the volume of revolution about the x-axis?
0.5 1
Integrating and substituting gives:

13. Example 2 What is the volume of revolution about the x-axis
for
between x = 1 and x = 4
Integrating gives:

15. What would be these Solids of Revolution about the x-axis?y x y
Torus
Sphere

16. What would be these Solids of Revolution about the x-axis?y x y
Torus
Sphere