1. Solids of Revolution Revolution about x-axis

2. What is a Solid of Revolution? Consider the area under the graph of from x = 0 to x = 2

3. What is a Solid of Revolution? If the shaded area is now rotated about the x-axis, then a three- dimensional solid (called Solid of Revolution) will be formed: http://chuwm2.tripod.com/revolution/Pictures from What will it look like?

4. What is a Solid of Revolution? 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. How is it calculated Consider the solid of revolution formed by the graph of y = x 2 from x = 0 to x = 2: What will it look like?

6. How is it calculated 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

7. How is it calculated xx x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

8. 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 sum of lots of circles … where area of circle = πr 2

9. 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 Example of a disk

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

11. Finding Volume

12. 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? Integrating and substituting gives: 0.5 1

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

14. Example 3 between x = 1 and x = 3 What is the volume of revolution about the x-axis Integrating gives: for

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

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

18. Disc Method:

19. What if the “slices” aren’t solid?

20. Washers Method

21. Washers Consider the area between two functions rotated about the axis Now we have a hollow solid We will sum the volumes of washers f(x) a b g(x)

22. Washers f(x) a b g(x) Outer Function Inner Function

23. The Method of Washers Find the volume of the solid formed by revolving the region bounded by y =  (x) and y = x² over the interval [0, 1] about the x – axis.

24. Solution:

25. Example #2 – Use Washer Method Find the volume of the solid formed by revolving the region bounded by and about the x-axis. Where do the curves meet?