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Unit 8: Applications of Integration
8.6 Finding the Volume of a Solid of Revolution Using the Shell Method
Setting up integrals using the shell method:
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In the disk/washer method, the axis of rotation is perpendicular to the axis of integration.
True
When rotating about the y-axis, the volume formula is V = 2\pi \int_{a}^{b} r(x) h(x) dx</latex>, where
r
(
x
)
r(x)
r
(
x
)
is the radius
The axis of revolution is critical for determining the direction of integration in the
shell method
.
True
What is the region being rotated in the example provided?
y
=
y =
y
=
x
\sqrt{x}
x
and
x
=
x =
x
=
4
4
4
What is the radius
r
(
y
)
r(y)
r
(
y
)
of the cylindrical shell when rotating the region bounded by
y
=
y =
y
=
x
\sqrt{x}
x
and
x
=
x =
x
=
4
4
4
about the x-axis?
y
y
y
For rotation about the y-axis, the volume integral is
V
=
V =
V
=
2
π
∫
a
b
r
(
x
)
h
(
x
)
d
x
2\pi \int_{a}^{b} r(x) h(x) dx
2
π
∫
a
b
r
(
x
)
h
(
x
)
d
x
, where
h
(
x
)
h(x)
h
(
x
)
is the height
For rotation about the x-axis, the volume integral is V = 2\pi \int_{c}^{d} r(y) h(y) dy</latex>, where
r
(
y
)
r(y)
r
(
y
)
is the radius
The disk/washer method uses slices perpendicular to the axis of rotation.
True
The axis of revolution determines the direction of integration in the
shell method
.
True
The shell method requires determining the limits of integration based on the bounds of the
region
being rotated.
True
What variable do the limits of integration represent when rotating about the y-axis and integrating with respect to x?
Bounds along the x-axis
The shell method is useful when the disk and washer methods become cumbersome or
impractical
Match the axis of revolution with its corresponding shell radius
y-axis ↔️
r
(
x
)
=
r(x) =
r
(
x
)
=
x
x
x
x-axis ↔️
r
(
y
)
=
r(y) =
r
(
y
)
=
y
y
y
The height function in the shell method is denoted by
h(x)
The radius function in the shell method is denoted by
r(x)
Match the axis of revolution with the correct height and radius functions:
y-axis ↔️
h
(
x
)
=
h(x) =
h
(
x
)
=
f
(
x
)
−
g
(
x
)
f(x) - g(x)
f
(
x
)
−
g
(
x
)
,
r
(
x
)
=
r(x) =
r
(
x
)
=
x
x
x
x-axis ↔️
h
(
y
)
=
h(y) =
h
(
y
)
=
f
(
y
)
−
g
(
y
)
f(y) - g(y)
f
(
y
)
−
g
(
y
)
,
r
(
y
)
=
r(y) =
r
(
y
)
=
y
y
y
The height function
h
(
x
)
h(x)
h
(
x
)
in the shell method represents the length of the cylindrical slice parallel to the axis of revolution.
True
Determining the correct limits of integration is essential when using the
shell method
.
True
What is the second step in determining the limits of integration for the shell method?
Determine range of variable
Match the axis of rotation with the correct limits of integration:
y-axis ↔️
2
π
∫
a
b
r
(
x
)
h
(
x
)
d
x
2\pi \int_{a}^{b} r(x) h(x) dx
2
π
∫
a
b
r
(
x
)
h
(
x
)
d
x
x-axis ↔️
2
π
∫
c
d
r
(
y
)
h
(
y
)
d
y
2\pi \int_{c}^{d} r(y) h(y) dy
2
π
∫
c
d
r
(
y
)
h
(
y
)
d
y
In the shell method, the axis of rotation is parallel to the
axis of integration
.
True
Steps to use the shell method for finding volumes
1️⃣ Choose the variable to integrate with respect to (x or y).
2️⃣ Determine the height
h
(
x
)
h(x)
h
(
x
)
or
h
(
y
)
h(y)
h
(
y
)
of the cylindrical shell.
3️⃣ Find the radius
r
(
x
)
r(x)
r
(
x
)
or
r
(
y
)
r(y)
r
(
y
)
from the axis of rotation to the shell.
4️⃣ Calculate the volume
V
V
V
using the appropriate integral.
What variable is used for integration when rotating about the y-axis in the example provided?
x
What is the radius
r
(
x
)
r(x)
r
(
x
)
of the cylindrical shell when rotating the region bounded by y = x^{2}</latex> and
y
=
y =
y
=
x
x
x
about the y-axis?
x
x
x
What is the height
h
(
y
)
h(y)
h
(
y
)
of the cylindrical shell when rotating the region bounded by
y
=
y =
y
=
x
\sqrt{x}
x
and
x
=
x =
x
=
4
4
4
about the x-axis?
4
−
y
2
4 - y^{2}
4
−
y
2
Match the volume calculation methods with their axis of rotation and slices:
Shell method ↔️ Parallel to axis, cylindrical shells
Disk/Washer method ↔️ Perpendicular to axis, disks or washers
The shell method is used when the axis of rotation is perpendicular to the axis of integration.
False
What is the radius
r
(
x
)
r(x)
r
(
x
)
of the cylindrical shell when rotating the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
x
x
x
about the y-axis?
x
x
x
The shell method is used when the axis of rotation is
parallel
to the axis of integration.
The radius function indicates the distance from the
axis
of revolution to the slice.
True
The height function
h
(
x
)
h(x)
h
(
x
)
or
h
(
y
)
h(y)
h
(
y
)
represents the length of the vertical or horizontal slice.
The limits of integration in the shell method depend on the bounds of the
region
being rotated.
Steps to determine the limits of integration for the shell method
1️⃣ Identify the region bounded by the given curves
2️⃣ Determine the range of the variable you are integrating with respect to (x or y)
3️⃣ Use this range to set the limits of integration
When rotating about the y-axis, the volume is given by V = 2\pi \int_{a}^{b} r(x) h(x) dx</latex>, where a and b represent the bounds along the
x-axis
Why is correctly identifying the limits of integration crucial in the shell method?
To set up the volume integral
What is the formula for calculating the volume when rotating about the y-axis using the shell method?
V
=
V =
V
=
2
π
∫
a
b
r
(
x
)
h
(
x
)
d
x
2\pi \int_{a}^{b} r(x) h(x) dx
2
π
∫
a
b
r
(
x
)
h
(
x
)
d
x
What does the axis of revolution determine in the shell method?
The direction of integration
What is the volume of the solid obtained by rotating the region bounded by
y
=
y =
y
=
x
\sqrt{x}
x
and
x
=
x =
x
=
4
4
4
about the x-axis?
8
π
8\pi
8
π
Match the axis of revolution with its corresponding radius function
y-axis ↔️
r
(
x
)
=
r(x) =
r
(
x
)
=
x
x
x
x-axis ↔️
r
(
y
)
=
r(y) =
r
(
y
)
=
y
y
y
What does the height function represent in the shell method?
Length of cylindrical slice
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