1. Shell Method for Volumes
Lesson 6-3a
Shell Method for Volumes

2. Quiz
AP Problem: The area of the region enclosed by the graphs of y = x and y = x² - 3x + 3 is
Reading questions: a) What is the area of a “shell”? b) If a disc or washer is perpendicular to the axis of rotation, then the shell is …. ?
4/3 sq units
A = 2πrh
parallel to the axis of rotation

3. Objectives
Find volume of revolved area solids using cylindrical shells

4. Vocabulary
None new

5. Volume = ∑ Area • thickness (∆variable)
Shell Method
An alternative to discs and washers is the shell method.
The cylinder is rolled out and represents the area.
Again the variable of integration can be either x or y.
Volume = ∑ Area • thickness (∆variable)
Sometimes it gives us an easy approach to solving the problem. dx
2πr h h
circumference dx
Volume = ∑ Cylinder Area • thickness = 2π ∫ r(x) h(x) dx
x = a
x = b

6. Comparison of Disk/Washers and Shell Methods
For the disc method:
Representative rectangle is always perpendicular to the axis of revolution
Rectangle represents the radius of the disc and its thickness
Area is circle (or washer) = πr² (or π[R² - r²])
For the shell method:
Representative rectangle is always parallel to the axis of revolution
Rectangle represents the height and thickness of the cylinder
Area is side of the cylinder rolled out = 2πrh
Usually r is either x or y and h is a function of the variable of integration

8. 6- 3 Example 1 ∆Volume = Area • Thickness Area = cylinders! = 2πrh
r = x
∆Volume = Area • Thickness
Area = cylinders! = 2πrh
= 2π (x) (x-½)
= 2π x1/2
Thickness = ∆x
X ranges from 1 out to 4
Area = L∙W
Volume = ∫ (2π x1/2) dx
x = 1
x = 4
= 2π ∫ (x1/2) dx
= 2π ((⅔)x3/2) |
= 2π (((⅔)(8)) – (2/3)) = (4π/3) (8 – 1)
= 28π/3 =
x = 1
x = 4 8

9. 6- 3 Example 2 ∆Volume = Area • Thickness Area = cylinders! = 2πrh
= 2π (y) (e-y²)
= 2π y e-y²
Thickness = ∆y
X ranges from 0 out to 1
Volume = ∫ (2π y e-y²) dy
y = 0
y = 1
let u = -y²,
so du = -2ydy
form: eu du
= 2π ∫ (y e-y²) dy
= -2π (e-y²) |
= -2π ((e-1) – (1)) = -2π (
= π = 3.972
y = 0
y = 1 9

10. Summary & Homework Summary: Homework:
Shell method is sometimes easier to use than either disc or washers
Discs or washers are always perpendicular to the axis of rotation
Shells are always parallel to the axis of rotation
Always use the method that makes the problem simpler to do
Homework:
pg 458 – 459 Day 1: 3, 6, 9, Day 2: 4, 7, 11, 38