1. Volumes – The Disk Method
Textbook 7.2
Overview on Page 456

2. Revolving a Function We have a function f(x) on the interval [a, b]
What happens when we revolve that segment of curve about the x axis?
What kind of functions generated these solids of revolution?
f(x) a b

3. Disks We can use integrals to determine the volume of these solids
f(x)
We can use integrals to determine the volume of these solids
Here we have a disk
which is a slice of the solid
What is the radius?
What is the thickness?
What then, is its volume? dx

4. Disks
To find the volume of the whole solid we sum the volumes of the disks
Shown as a definite integral
f(x) a b

5. Try It Out!
Try the function y = x3 on the interval 0 < x < 2 rotated about x-axis

6. Revolve About Line Not a Coordinate Axis
Consider the function y = 2x2 and the boundary lines y = 0, x = 2
Revolve this region about the line x = 2
We need an expression for the radius in terms of y

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

8. What will be the limits of integration?
Application
Given two functions y = x2, and y = x3
Revolve region between about x-axis
What will be the limits of integration?

9. Revolving About y-Axis
Also possible to revolve a function about the y-axis
Make a disk or a washer to be horizontal
Consider revolving a parabola about the y-axis
How to represent the radius?
What is the thickness of the disk?

10. Revolving About y-Axis
Must consider curve as x = f(y)
Radius = f(y)
Slice is dy thick
Volume of the solid rotated about y-axis

11. Flat Washer
Determine the volume of the solid generated by the region between y = x2 and y = 4x, revolved about the y-axis
Radius of inner circle?
f(y) = y/4
Radius of outer circle?
Limits?
0 < y < 16