FIND THE VOLUME OF REVOLUTION USING THE DISK METHOD

FIND THE VOLUME OF REVOLUTION USING THE WASHER METHOD

FIND THE VOLUME OF A SOLID WITH KNOWN CROSS SECTIONS

7.2a Volume: The Disk Method

Area is only one of the applications of integration. We can add up representative volumes in the same way we add up representative rectangles.When we are measuring volumes of revolution, we can slice representative disks or washers.

These are a few of the many industrial uses for volumes of revolutions.

If a region in the plane is revolved about a line, it creates a solid of revolution. The line is called the axis of revolution. The simplest solid would be a right circular cylinder, called a disk. It’s volume would be the area of the circular base, times the height, which in this case is the width of the disk. Slice a bunch of these off, add up the volumes, and you get the volume of the entire solid.

2V R x 2( )iV R x x

f(∆x)=

The Disk MethodTo find the volume of a solid of revolution with the disk method, use one of the following.

2Volume [ ( )]b

a

V R x dx Horizontal Axis of Revolution

y=

Vertical Axis of Revolution

2Volume [ ( )]d

c

V R y dy

=x

Ex 1 p. 458 Using the disk methodFind the volume of the solid formed by revolving the region bounded by the graph of ( ) , the line x = 0, x= 2 and the x-axis, about the x-axisf x x

For a 3-D look, click here http://www.ies.co.jp/math/java/calc/rotate/rotate.html

Each disk is a cylinder with width ∆x and radius of the circular base of f(x)Integration allows you to take each of these volumes and add them up, allowing the radius to change as the f(x) value changes.2[ ( )]

b

a

f x dx2

2

0

[ ]x dxLowest x

Highest xRadius = y in terms of

x

∆x

22

0( ) (2 0) 22x

2

0

xdx

Ex 2 p.458 Revolving about a line that is not a coordinate axis.Find the volume of the solid formed by revolving the solid formed by 2( ) 3 and ( ) 2f x x g x about the line y = 2

Here are some questions to be answered:Where do the two functions intersect?

What is the radius at any given x-value?

How are you stacking things? Along x-axis or along y-axis?

If you are stacking things along the x-axis, what is the lowest x value? What is the highest x-value?

122

1

(3 ) 2x dx

1

22

1

1 x dx

23 2x

( ) ( ) ( )R x f x g x 2( ) 3 2R x x

Along x-axis

x = -1, 1

16

15

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7-8

Figure 7.19

Let’s look at a volume of revolution with a hole in the middle. This is called the “washer method” because each representative slice looks like a washer.

To visualize, click here.http://mathdemos.org/mathdemos/washermethod/gallery/gallery.html

Let’s look at a pretty basic volume. Find the volume of rotation formed by the line y = 2x, the line y = 1, from x = 1/2 to x = 3, about the x-axis

6

4

2

-2

2 2V R r x

2 2( )R r x

2 2( )b

a

R r dx

Which becomes R = 2x

r = 1

Find the volume of rotation formed by the line y = 2x, the line y = 1, from x = 1/2 to x = 3, about the x-axis

6

4

2

-2

2 2( )b

a

R r dx

3

2 2

12

( 2 1 )x dx 3

2

12

(4 1)x dx

100

3

These can rotate around the y-axis instead, and then you would stack upward instead of to the right.

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region y = x2 between the y-axis, the line y=4, and graph, revolving about the y-axis

4

2

2( )

d

c

f y dyI need a function in terms of y for the radius, the lowest value of y and the highest value of y.

2y xy x

42

0

y dy 4

0

ydy 8

7.2a p. 463/ 1-15 all

http://www.math.psu.edu/dlittle/java/calculus/volumedisks.html

http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html