1. Volumes
Lesson 6.2

2. 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
f(x) a c b

3. Cross Sections
f(x)
This suggests a limit and an integral a c b

4. Cross Sections
We could do similar summations (integrals) for other shapes
Triangles
Semi-circles
Trapezoids
f(x) a c b

5. 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

6. 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

7. 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

8. 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

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

10. 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

11. 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?

12. 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?

13. 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

14. 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

15. Assignment
Lesson 6.2A
Page 372
Exercises 1 – 29 odd

16. Find the volume generated when this shape is revolved about the y axis.
We can’t solve for x, so we can’t use a horizontal slice directly.

17. If we take a vertical slice
and revolve it about the y-axis
we get a cylinder.
This model of the shell method and other calculus models are available from: Foster Manufacturing Company, 1504 Armstrong Drive, Plano, Texas Phone/FAX: (972)

18. Shell Method Based on finding volume of cylindrical shells
Add these volumes to get the total volume
Dimensions of the shell
Radius of the shell
Thickness of the shell
Height

19. The Shell Consider the shell as one of many of a solid of revolution
The volume of the solid made of the sum of the shells dx
f(x)
f(x) – g(x) x
g(x)

20. Try It Out!
Consider the region bounded by x = 0, y = 0, and

21. Hints for Shell Method Sketch the graph over the limits of integration
Draw a typical shell parallel to the axis of revolution
Determine radius, height, thickness of shell
Volume of typical shell
Use integration formula

22. Rotation About x-Axis
Rotate the region bounded by y = 4x and y = x2 about the x-axis
What are the dimensions needed?
radius
height
thickness
thickness = dy
radius = y

23. Rotation About Non-coordinate Axis
Possible to rotate a region around any line
Rely on the basic concept behind the shell method
f(x)
g(x)
x = a

24. Rotation About Non-coordinate Axis
What is the radius?
What is the height?
What are the limits?
The integral: r
f(x)
g(x)
a – x
x = c
x = a
f(x) – g(x)
c < x < a

25. Try It Out
Rotate the region bounded by 4 – x2 , x = 0 and, y = 0 about the line x = 2
Determine radius, height, limits
r = 2 - x
4 – x2

26. Try It Out
Integral for the volume is

27. Assignment
Lesson 6.2B
Page 373
Exercises 41 – 59 odd