Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

Disks and Washers: Using Integration to Find Volume

Limerick Nuclear Generating Station, Pottstown, Pennsylvania

y x Suppose I start with this curve.

My boss at the ACME Rocket Company has assigned me to build a cone in this shape.

So I put a piece of wood in a lathe and turn it to a shape to match the curve.

Find the volume of the solid created when the graph is rotated about the x-axis.

y x

y xHow could we find the volume of the cone?

One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.

The volume of each flat cylinder (disk) is:

2 the thicknessr

In this case:

r= the y value of the function

thickness = a small change

in x = dx

2

x dx

y xThe volume of each flat cylinder (disk) is:

2 the thicknessr

If we add the volumes, we get:

24

0x dx

4

0 x dx

8

2

x dx

4

0xdx

Example• Find the volume of the solid that results

when the region enclosed by

is revolved about the x-axis.

3 ,0 0,y ,2 xxxy

Example• Find the volume of the solid that results

when the region enclosed by

is revolved about the y-axis 41 ,

1 y

yx

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis:

2.000574 .439 185x y y x

y

500 ft

500 22

0.000574 .439 185 y y dy

The volume can be calculated using the disk method with a horizontal disk.

324,700,000 ft

This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.

The washer method formula is: 2 2 b

aV R r dx

Example• Find the volume of the solid that results

when the region enclosed by

is revolved about the x-axis.

4 ,1 ,1 , xxyxy

Example• Find the volume of the solid that results

when the region enclosed by

is revolved about the y-axis.

4 ,1 ,1 , xxyxy

Example• Find the volume of the solid that results

when the region enclosed by

is revolved about the y-axis.

xyxy 2 and 2

Example• Find the volume of the solid that results

when the region enclosed by

is revolved about the line x = 2.

xyxy 2 and 2

Practice

• Pg. 456• 1, 3, 7, 9, 19, 23