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Finding volumes of solids of revolution using the washer method.

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Often, the volume of a solid of revolution can be found using the disk method. However, if a region is revolved about a line that does not touch or pass through it, the resulting solid of revolution will have a hole in it. In this case, the volume of the solid should be found using a generalization of the disk method called the washer method.

Many objects can be viewed as solids of revolution. When the object has a cylindrical hole in it, the washer method is often a good way to find the object’s volume.

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The Large Hadron Collider can be viewed as a solid of revolution, and its volume can be found using the washer method. The engineers who planned the structure needed to know the volume to make sure conditions inside it could be regulated appropriately.
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Many beads can be considered solids of revolution whose volumes can be found using the washer method. Manufacturers can use this to know what volume of material would be needed for an order.
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Many pendant lamps are solids of revolution. During the design of such a lamp, an artist would need to know the volume of material in the shade. That volume can be found using the washer method.

Volumes of solids of revolution - Washer
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Revolve Region
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# of washers

r :

R :

Thickness of each washer:  0.50
Sum of volumes of washers:  320.08
Volume of torus:
∫ − 2 2 16π 4 − y 2  dy≈315.83
# of washers

h :

r :

a = 3.50
Thickness of each washer:  0.50
Sum of volumes of washers:  180.05
Volume of bead:
∫−3.503.50π(12.52−x2)dx ≈179.59
# of washers

a :

b :

c :

Thickness of each washer:  0.50
Sum of volumes of washers:  127.104
Volume of solid: ∫ 04.5 π[(−0.67x+5 ) 2−4]dx≈127.235

INSTRUCTIONS

When a circle is revolved about a line that does not pass through it, the resulting solid is called a torus. Change the values of the parameters to see how the graph changes. Revolve the rectangles about the y-axis to form a solid made of washers. As the number of rectangles and washers increases, the solid approximates a torus more and more closely.
Click and drag anywhere in the viewing window to inspect the region more closely. Click the reset button to return the figure to its default position.

PRACTICE QUESTIONS

1 2 3
1. To two decimal places, what is the volume of the torus when R=4 and
r = 2 ?
2. When there are 16 washers with R = 5 and r = 2, what is the total volume of the washers? Round to two decimal places.
3. What happens as the number of washers increases? Select all that apply.

INSTRUCTIONS

If a chord of a circle centered at the origin is horizontal, revolving the sector it cuts off about the x-axis produces a sphere with a hole bored through it. This solid resembles a spherical bead.
Click and drag anywhere in the viewing window to inspect the region more closely. Click the reset button to return the figure to its default position.

PRACTICE QUESTIONS

1 2 3
1. To two decimal places, what is the volume of the bead when r = 4 and
h = 1 ?
2. When r = 3 and h = 0.5, how many washers are needed for the thickness of each washer to be about 0.59 ?
3. If the volume of the bead is not zero, it can be found using disks rather than washers if which of the following is true? Select all that apply.

INSTRUCTIONS

If a right triangle has a vertical leg on the positive
y-axis and a horizontal leg in quadrant I, then revolving the triangle about the x-axis produces a cone with a hole drilled through it. If the horizontal leg of the triangle is on the x-axis, then revolving the triangle about the x-axis produces a cone.
Click and drag anywhere in the viewing window to inspect the region more closely. Click the reset button to return the figure to its default position.

PRACTICE QUESTIONS

1 2 3
1. To three decimal places, what is the volume of the solid if a = 2.5, b = 4, and c = 3 ?
2. Let a = 3, b = 4 and
c = 2. What is the minimum number of washers needed for the approximate volume to match the exact volume to three decimal places?
3. If the volume of the solid is not zero, it can be found using disks rather than washers if which of the following is true? Select all that apply.
Result

x can be any number from to