1. Volumes by Disks and Washers
Or, how much toilet paper fits on one of those huge rolls, anyway??
Adapted from Howard Lee
8 June 2000 & Greg Kelly

2. A Real Life Situation Relief
That’s a lotta toilet paper! I wonder how much is actually on that roll?

3. How do we get the answer? CALCULUS!!!!!
(More specifically: Volumes by Integrals)

4. Volume by Slicing Volume = length x width x height
Volume of a slice = Area of a slice x Thickness of a slice A t
Total volume =  (A x t)

5. Volume by Slicing VOLUME =  A dt Total volume =  (A x t)
But as we let the slices get infinitely thin,
Volume = lim  (A x t)
t  0
VOLUME =  A dt
Recall: A = area of a slice

6. Rotating a Function x=f(y) x=f(y)
Such a rotation traces out a solid shape
(in this case, we get something like half an egg)

7. Volume by Slices
} dt
Slice r
Thus, the area of a slice is r2
A = r2

8. Disk Formula VOLUME =  A dt VOLUME =   r2dt But: A =  r2, so…
“The Disk Formula”

9. Volume by Disks VOLUME =   f(y) 2dy y axis Slice x = f(y) r x dy
} thickness x
radius
x axis
Thus, A = x2
but x = f(y)
and dt = dy, so...
VOLUME =   f(y) 2dy

10. More Volumes Slice f(x) g(x) Area of a slice = (R2-r2) dt R r
rotate around x axis
Area of a slice = (R2-r2) dt

11. Washer Formula VOLUME =  A dt VOLUME =  (R2 - r2) dt
But: A =  (R2 - r2), so…
VOLUME =  (R2 - r2) dt
“The Washer Formula”

12. Volumes by Washers V =  (f(x) 2- g(x) 2) dx Slice f(x) g(x) dt dx
Big R
f(x) R
g(x)
little r r
g(x) dt dx
Thus, A = (R2 - r2)
= (f(x) 2- g(x) 2)
V =  (f(x) 2- g(x) 2) dx

13. The application we’ve been waiting for...
f(x)
g(x) 2 1
0.5
rotate around x axis 1

14. Toilet Paper V =  (f(x) 2- g(x) 2) dx
So we see that:
f(x) = 2, g(x) = 0.5 1
g(x)
0.5 1
V =  (f(x) 2- g(x) 2) dx
x only goes from 0 to 1,
so we use these as the limits of integration.
Now, plugging in our values for f and g:
V =  (22 - (0.5) 2) dx = 3.75 (1 - 0) =
3.75  1

15. Suppose I start with this curve.
My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.
So I put a piece of wood in a lathe and turn it to a shape to match the curve.

16. 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 thin slices (flat cylinders) and add their volumes.
The volume of each flat cylinder (disk) is:
In this case:
r= the y value of the function
thickness = a small change in x = dx

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

18. This application of the method of slicing is called the disk method
This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.
If the shape is rotated about the x-axis, then the formula is:
Since we will be using the disk method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.
A shape rotated about the y-axis would be:

19. y-axis is revolved about the y-axis. Find the volume.
The region between the curve , and the
y-axis is revolved about the y-axis. Find the volume. y x
We use a horizontal disk.
The thickness is dy.
The radius is the x value of the function
volume of disk

20. The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis:
The volume can be calculated using the disk method with a horizontal disk.

21. and is revolved about the y-axis. Find the volume.
The region bounded by
and is revolved about the y-axis.
Find the volume.
If we use a horizontal slice:
The “disk” now has a hole in it, making it a “washer”.
The volume of the washer is:
outer
radius
inner
radius

22. This application of the method of slicing is called the washer method
This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.
The washer method formula is:

23. p If the same region is rotated about the line x=2:
The outer radius is:
The inner radius is: r R p