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Volumes of Revolution Disks and Washers

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1 Volumes of Revolution Disks and Washers
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 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

4 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

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

6 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

7 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?

8 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?

9 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

10 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

11 Assignment Lesson 7.2 Page 272 Exercises 1 – 23 odd


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