Volumes of solids of revolution

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Finding volumes of solids of revolution using the washer method.

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.

Volumes of solids of revolution - Washer

EQUATIONS

Revolve Region

Scenario 1

Scenario 2

Scenario 3

# of washers

n11020304050

r :

R :

Thickness of each washer:

$0.50$

Sum of volumes of washers:

$320.08$

Volume of torus:

$\int_{- 2}^{2} 16 π \sqrt{4 - y^{2}}$

$d y \approx 315.83$

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. To two decimal places, what is the volume of the torus when

$R = 4$ and

$r = 2$ ?

$100.53$

$157.91$

$201.06$

$315.83$

$631.65$

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.

$394.78$

$395.13$

$396.68$

$400.10$

$409.58$

3. What happens as the number of washers increases? Select all that apply.

The thickness of each washer decreases.

The thickness of each washer increases.

The sum of the volumes of the washers decreases.

The sum of the volumes of the washers increases.

The sum of the volumes of the washers approaches the volume of the torus.

The size of the torus increases.

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. To two decimal places, what is the volume of the bead when

$r = 4$ and

$h = 1$ ?

$94.78$

$95.02$

$243.35$

$243.97$

$403.10$

$404.13$

2. When

$r = 3$ and

$h = 0.5,$ how many washers are needed for the thickness of each washer to be about

$0.59$ ?

$5$

$6$

$8$

$10$

$12$

$14$

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.

$r = 0$

$r > h$

$r = h$

$h = 0$

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. To three decimal places, what is the volume of the solid if

$a = 2.5,$

$b = 4,$ and

$c = 3$ ?

$16.755$

$26.180$

$42.390$

$42.412$

$66.759$

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?

$30$

$35$

$40$

$45$

$50$

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.

$a = 0$

$b = 0$

$c = 0$

$a = b$

$a = c$

$b = c$

$a < c$

$a > c$

Result

$x$ can be any number from to

YOU WILL LEARN ABOUT:

INSTRUCTIONS

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

INSTRUCTIONS

If a chord of a circle centered at the origin is horizontal, revolving the sector it cuts off about the x<math xmlns="http://www.w3.org/1998/Math/MathML" mathsize="15px" class=""><mi>x</mi></math>-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

INSTRUCTIONS

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

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.