Save
...
Unit 8: Applications of Integration
8.5 Finding the Volume of a Solid of Revolution Using the Washer Method
Setting up integrals using the washer method:
Save
Share
Learn
Content
Leaderboard
Share
Learn
Cards (42)
What is the first step in setting up an integral using the washer method?
Identify the region
What is the formula for the volume when revolving around the x-axis?
\int_{a}^{b} \pi [R(x)^{2} - r(x)^{2}] \, dx</latex>
Before setting up an integral using the washer method, it is essential to clearly identify the region to be
rotated
To use the washer method, it is essential to understand the functions defining the region and the axis of
revolution
What determines the setup of the volume integral in the washer method?
Axis of revolution
The function defining the upper or rightmost boundary of the region being rotated is used to find the
outer
radius.
When the axis of rotation is shifted, offsets must be included to adjust the
radius
.
If the region is bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
, the outer radius when rotated about the x-axis is
R
(
x
)
=
R(x) =
R
(
x
)
=
4
4
4
.
True
The upper boundary of a region rotated about the x-axis forms the
outer radius
.
True
What integral is used when revolving around the y-axis using the washer method?
\int_{c}^{d} \pi [R(y)^{2} - r(y)^{2}] \, dy</latex>
Steps to set up the integral for volume using the washer formula when revolving around the x-axis
1️⃣ Determine the axis of revolution
2️⃣ Find the outer radius function
3️⃣ Find the inner radius function
4️⃣ Set up the integral using dx
What are the two primary orientations for the axis of revolution in the washer method?
x-axis and y-axis
When revolving around the x-axis, the integrals are set up with respect to x</latex> using
d
x
dx
d
x
True
When revolving around the y-axis, the integrals are set up with respect to
y
Match the axis of revolution with its corresponding setup:
x-axis ↔️
∫
a
b
π
[
R
(
x
)
2
−
r
(
x
)
2
]
d
x
\int_{a}^{b} \pi [R(x)^{2} - r(x)^{2}] \, dx
∫
a
b
π
[
R
(
x
)
2
−
r
(
x
)
2
]
d
x
y-axis ↔️
∫
c
d
π
[
R
(
y
)
2
−
r
(
y
)
2
]
d
y
\int_{c}^{d} \pi [R(y)^{2} - r(y)^{2}] \, dy
∫
c
d
π
[
R
(
y
)
2
−
r
(
y
)
2
]
d
y
What region is formed when
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
are rotated about the x-axis?
Area enclosed by functions
Steps to find the outer radius of a washer
1️⃣ Identify the function farthest from the axis
2️⃣ Express in terms of the appropriate variable
3️⃣ Include any offsets if needed
If the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
is rotated about the x-axis, the outer radius is
R
(
x
)
=
R(x) =
R
(
x
)
=
4
4
4
True
What is the first step in finding the inner radius of a washer?
Identify the function closest
Which function defines the lower or leftmost boundary of the region being rotated?
The function closest to the axis
If the axis of revolution is the y-axis, the inner radius function is
r
(
y
)
=
r(y) =
r
(
y
)
=
g
(
y
)
g(y)
g
(
y
)
.
True
What is the inner radius when the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
is rotated about the x-axis?
r
(
x
)
=
r(x) =
r
(
x
)
=
x
2
x^{2}
x
2
Steps to find the inner radius of a washer in the washer method
1️⃣ Identify the function closest to the axis
2️⃣ Express in terms of the appropriate variable
3️⃣ Include any offsets if the axis is shifted
Which function defines the lower or leftmost boundary of the region being rotated?
The function closest to the axis
When revolving around the x-axis, the integrals are set up with respect to x using
d
x
dx
d
x
.
When revolving around the y-axis, the radius functions are
R
(
y
)
R(y)
R
(
y
)
and
r
(
y
)
r(y)
r
(
y
)
.
True
What is the axis of revolution in the washer method used to determine?
Volume integral setup
When revolving around the y-axis, the
integrals
are set up with respect to y.
True
What is the formula for the outer radius when revolving around the x-axis?
R(x) = f(x)</latex>
What is the first step to find the inner radius of a washer?
Identify the function closest to the axis
What is the first step to find the outer radius of a washer in the washer method?
Identify the function farthest from the axis
When rotating about the x-axis, the outer radius is expressed as
R
(
x
)
=
R(x) =
R
(
x
)
=
f
(
x
)
f(x)
f
(
x
)
.
True
What variable is used to express the outer radius when rotating about the y-axis?
y
Steps to find the inner radius of a washer in the washer method
1️⃣ Identify the function closest to the axis
2️⃣ Express in terms of the appropriate variable
3️⃣ Include any offsets if the axis is shifted
When rotating about the x-axis, the inner radius is expressed as
r
(
x
)
=
r(x) =
r
(
x
)
=
f
(
x
)
f(x)
f
(
x
)
using the variable x.
Match the boundary role with its corresponding function:
Lower boundary ↔️ Forms the inner radius
Upper boundary ↔️ Forms the outer radius
Axis of rotation ↔️ Determines the axis of revolution
When revolving around the x-axis, the integrals are set up with respect to
x
What is the integral formula for the volume when revolving around the x-axis?
∫
a
b
π
[
R
(
x
)
2
−
r
(
x
)
2
]
d
x
\int_{a}^{b} \pi [R(x)^{2} - r(x)^{2}] \, dx
∫
a
b
π
[
R
(
x
)
2
−
r
(
x
)
2
]
d
x
To find the outer radius, identify the function farthest from the
axis
If the region is bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
and rotated about the x-axis, the outer radius is
R
(
x
)
=
R(x) =
R
(
x
)
=
4
4
4
.
True
See all 42 cards