1. Volumes – The Disk Method
Lesson 7.2

2. Revolving a Function Consider a function f(x) on the interval [a, b]
Now consider revolving that segment of curve about the x axis
What kind of functions generated these solids of revolution?
f(x) a b

3. Revolving a Function
What kind of function here?

4. Disks
f(x)
We seek ways of using integrals to determine the volume of these solids
Consider a disk which is a slice of the solid
What is the radius
What is the thickness
What then, is its volume? dx

5. 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

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

7. 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

8. 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

9. 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?

10. 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?

11. 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

12. 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

13. Assignment A Lesson 7.2A Page 463 Exercises 1 – 29 odd
Spreadsheet Assignment Due in 1 Week

14. Cross Sections
Consider a square at x = c with side equal to side s = f(c)
Now let this be a thin slab with thickness Δx
What is the volume of the slab?
Now sum the volumes of all such slabs

15. Cross Sections
This suggests a limit and an integral

16. Cross Sections
We could do similar summations (integrals) for other shapes
Triangles
Semi-circles
Trapezoids

17. Try It Out Consider the region bounded
above by y = cos x
below by y = sin x
on the left by the y-axis
Now let there be slices of equilateral triangles erected on each cross section perpendicular to the x-axis
Find the volume

18. Assignment
Lesson 7.2B
Page 464
Exercises odd, 49, 53, 57