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AP Calculus BC
Unit 8: Applications of Integration
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
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The Washer Method involves integrating the area of circular washers perpendicular to the axis of
revolution
The Washer Method formula uses the difference of squares of the outer and
inner
radii to calculate volume.
What does
R
(
x
)
R(x)
R
(
x
)
represent in the Washer Method formula?
Outer radius
Steps to apply the Washer Method
1️⃣ Sketch the region between the outer and inner curves
2️⃣ Determine the outer and inner radii
3️⃣ Set up the integral for volume
4️⃣ Evaluate the integral to find the volume
What is the volume of the solid generated by revolving the region between
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
x
x
x
from
x
=
x =
x
=
0
0
0
to
x
=
x =
x
=
1
1
1
around the x-axis?
2
π
15
\frac{2\pi}{15}
15
2
π
The Disk Method is used when revolving a region between two curves around an axis.
False
What is the volume of the solid generated by revolving
y
=
y =
y
=
x
\sqrt{x}
x
from
x
=
x =
x
=
0
0
0
to
x
=
x =
x
=
4
4
4
around the x-axis using the Disk Method?
8
π
8\pi
8
π
The Washer Method is used for regions between two
curves
What does
r
(
x
)
r(x)
r
(
x
)
represent in the Washer Method formula?
Inner radius
The Disk Method formula includes the subtraction of two squared radii.
False
What is the first step in identifying the axis of revolution?
Sketch the region
What is the Washer Method used to calculate?
Volume of a solid
Match the axis of revolution with its description:
x-axis ↔️ Region rotated around the x-axis
y-axis ↔️ Region rotated around the y-axis
What is the axis of revolution in the Washer Method?
The line around which a region is rotated
Common axes of revolution are the x-axis and the
y-axis
.
Revolving a region around the y-axis requires expressing curves in terms of
y
What is the Washer Method formula when revolving around the x-axis?
V = \pi \int_{a}^{b} (R(x)^{2} - r(x)^{2}) \, dx</latex>
Steps to determine outer and inner radii in the Washer Method
1️⃣ Sketch the region
2️⃣ Identify the axis of revolution
3️⃣ Express radii in terms of
x
x
x
or
y
y
y
4️⃣ Determine outer radius
R
(
x
)
R(x)
R
(
x
)
5️⃣ Determine inner radius
r
(
x
)
r(x)
r
(
x
)
What is the outer radius
R
(
x
)
R(x)
R
(
x
)
in the Washer Method?
Distance from axis to farther function
What is the inner radius
r
(
x
)
r(x)
r
(
x
)
in the Washer Method?
Distance from axis to closer function
When revolving around the x-axis, the Washer Method formula uses
R
(
x
)
R(x)
R
(
x
)
and
r
(
x
)
r(x)
r
(
x
)
.
Match the radii with the correct functions when revolving around the x-axis:
y
=
y =
y
=
x
x
x
↔️
R
(
x
)
=
R(x) =
R
(
x
)
=
x
x
x
y
=
y =
y
=
x
2
x^{2}
x
2
↔️
r
(
x
)
=
r(x) =
r
(
x
)
=
x
2
x^{2}
x
2
What is the final Washer Method integral for revolving between
y
=
y =
y
=
x
x
x
and
y
=
y =
y
=
x
2
x^{2}
x
2
around the x-axis?
V
=
V =
V
=
π
∫
0
1
(
x
2
−
x
4
)
d
x
\pi \int_{0}^{1} (x^{2} - x^{4}) \, dx
π
∫
0
1
(
x
2
−
x
4
)
d
x
The outer and inner radii in the Washer Method are always measured from the
axis of revolution
.
The Washer Method is used to calculate the
volume
of a solid of revolution.
What does
R
(
x
)
R(x)
R
(
x
)
represent in the Washer Method formula?
Outer radius
What does r(x)</latex> represent in the Washer Method formula?
Inner radius
The Washer Method involves integrating the area of circular washers perpendicular to the
axis
of revolution.
The volume of the solid formed by revolving between y = x^{2}</latex> and
y
=
y =
y
=
x
x
x
around the x-axis from
x
=
x =
x
=
0
0
0
to
x
=
x =
x
=
1
1
1
is
2
π
15
\frac{2\pi}{15}
15
2
π
What is the Washer Method used to calculate?
Volume of solids of revolution
The Washer Method involves integrating the area of circular washers perpendicular to the axis of
revolution
The Washer Method formula includes the difference between the squares of the outer and inner
radii
.
What does
R
(
x
)
R(x)
R
(
x
)
represent in the Washer Method formula?
Outer radius
What does
r
(
x
)
r(x)
r
(
x
)
represent in the Washer Method formula?
Inner radius
The interval of integration in the Washer Method is denoted by
[a, b]
Steps to apply the Washer Method
1️⃣ Sketch the region and identify the outer and inner curves
2️⃣ Determine
R
(
x
)
R(x)
R
(
x
)
and
r
(
x
)
r(x)
r
(
x
)
in terms of
x
x
x
3️⃣ Set up the integral to calculate the volume
4️⃣ Evaluate the integral
In the example,
R
(
x
)
=
R(x) =
R
(
x
)
=
x
x
x
and
r
(
x
)
=
r(x) =
r
(
x
)
=
x
2
x^{2}
x
2
.
What is the volume of the solid generated by revolving the region between
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
x
x
x
from
x
=
x =
x
=
0
0
0
to
x
=
x =
x
=
1
1
1
around the x-axis?
2
π
15
\frac{2\pi}{15}
15
2
π
What are the two methods for calculating the volume of solids of revolution?
Disk and Washer Methods
The Disk Method uses the formula V = \pi \int_{a}^{b} [f(x)]^{2} \,
dx
</latex>.
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