1. Solids of Revolution Washer Method
Section 7-2 cont
Solids of Revolution
Washer Method

2. 1. What is the volume of the hollow cylinder?

3. The Washer Method
The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a representative washer.
The washer is formed by revolving a rectangle about an axis, as shown.
If r and R are the inner and outer radii
of the washer and w is the width of the washer, the volume is given by
Volume of washer = π(R2 – r2)w.

4. The Washer Method
To see how this concept can be used to find the volume of a solid of revolution, consider a region bounded by an outer radius R(x) and an inner radius r(x), as shown in Figure 7.19.
Figure 7.19

5. Washer method
If the area between two curves is revolved around an axis, a solid is created that is “hollow” in the center.
When slicing this solid, the sections created are washers, not solid disks.
The area of the smaller circle must be subtracted from the area of the larger one.

6. Volume of Washer: Horizontal axis of revolution
Slice with rectangle perpendicular to axis
Integrate in terms of x
Vertical axis of revolution
Slice with rectangles perpendicular to axis
Integrate in terms of y
Width – w
Outer Radius (R) –
top function
Inner Radius (r) –
bottom function
Right most
function
Left most
function

7. 6) Find the volume of the region bounded by y = x and y = x2 from x = 0 to x = 1 revolved about the x – axis.
Outer radius – R
Inner radius – r

8. 7) Find the volume of the region bounded by x = y2 and x = y3 from y = 0 to y = 1 revolved about x = 0 (y-axis) 1
x = y2
x = y3

9. Are regions always revolved about the x or y axis?
Axes of revolutions can be lines as well
Radii need adjusted
x = -2
For vertical slices x = number
For horizontal slices y = number
Formulas stay the same
Radii need adjusted
y = 5
R = 5 – x
R = y + 2
There are two more situations each is similar to these two

10. 8) Find the volume of the region bounded by y = x2 and y = 4x revolved around the line y = -2
Outer Radius f(x) + 2 4
y = -2
Inner Radius g(x) + 2

11. 3) Continued

12. Assignment
Worksheet 7-2 Washers