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AP Calculus BC
Unit 8: Applications of Integration
8.14 Volume with Shell Method: Revolving Around Other Horizontal or Vertical Lines
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What is the Shell Method used to calculate?
Volume of 3D solids
What is the Axis of Revolution in the Shell Method?
Line around which region revolves
What does the Shell Method slice the region into?
Cylindrical shells
In the Disk Method, the axis of revolution is typically the
x-axis
or y-axis.
What is the radius in the Shell Method for a cylindrical shell?
Distance from axis of revolution
When revolving around a horizontal axis, you integrate with respect to
y
y
y
.
True
When revolving around a vertical axis, the height is a function of
y
y
y
.
False
When revolving around a horizontal axis, the height is a function of
x
x
x
.
False
To determine the limits of integration, you need to find the points where the bounding functions
intersect
The limits of integration are denoted as
a
a
a
and
b
b
b
when revolving around a vertical axis
The Shell Method is particularly useful when the
Disk Method
becomes complicated.
True
The Shell Method can be used for any horizontal or vertical
axis
of revolution.
True
Match the feature with the correct method:
Axis aligned with bounding functions ↔️ Disk Method
Axis not aligned with bounding functions ↔️ Shell Method
The axis of revolution determines the shape of the 3D solid.
True
The height in the Shell Method is measured along the axis of revolution.
False
What method is used to calculate volume when revolving regions around axes other than the coordinate axes?
Shell Method
The radius in the Shell Method is the distance from the axis of revolution to the edge of the
shell
The height of a cylindrical shell is measured along the axis perpendicular to the
axis of revolution
.
True
What is the integral formula for calculating volume when revolving around a horizontal axis using the Shell Method?
V
=
V =
V
=
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
If the region bounded by
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
and g(x) = 4x</latex> is revolved around
x
=
x =
x
=
−
1
- 1
−
1
, the radius
r
(
x
)
r(x)
r
(
x
)
is given by x + 1
To evaluate the integral in the Shell Method, we apply the formula
V = 2\pi \int_{a}^{b} r(x)h(x) dx
Match the components of the Shell Method formula with their definitions:
r
(
x
)
r(x)
r
(
x
)
or
r
(
y
)
r(y)
r
(
y
)
↔️ Radius from the axis of revolution
h
(
x
)
h(x)
h
(
x
)
or
h
(
y
)
h(y)
h
(
y
)
↔️ Height of the cylindrical shells
a
,
b
a, b
a
,
b
or
c
,
d
c, d
c
,
d
↔️ Limits of integration
When revolving around a horizontal axis
y
=
y =
y
=
k
k
k
, the radius
r
(
y
)
r(y)
r
(
y
)
is given by y - k</latex>.
True
For a vertical axis of revolution
x
=
x =
x
=
h
h
h
, the radius
r
(
x
)
r(x)
r
(
x
)
is x - h
The Shell Method is useful when revolving around axes other than the x-axis or
y-axis
.
True
The Shell Method can be used with any horizontal or vertical line as the
axis of revolution
.
True
Match the method with its characteristic:
Shell Method ↔️ Revolves around any axis
Disk Method ↔️ Revolves around x or y-axis
Volume ↔️
V
=
V =
V
=
2
π
∫
a
b
f
(
x
)
g
(
x
)
d
x
2\pi \int_{a}^{b} f(x)g(x) dx
2
π
∫
a
b
f
(
x
)
g
(
x
)
d
x
The Shell Method allows for revolving around any horizontal or vertical line, unlike the
Disk Method
.
True
The radius in the Shell Method is the distance from the axis of revolution to the edge of the
shell
The height in the Shell Method is the length of the shell along the axis
perpendicular
Match the axis of revolution with the correct integration variable:
Horizontal ↔️
y
y
y
Vertical ↔️
x
x
x
When revolving the region bounded by
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
and
g
(
x
)
=
g(x) =
g
(
x
)
=
4
x
4x
4
x
around
x
=
x =
x
=
−
1
- 1
−
1
, the radius
r
(
x
)
=
r(x) =
r
(
x
)
=
x
+
x +
x
+
1
1
1
and the height
h
(
x
)
=
h(x) =
h
(
x
)
=
4
x
−
x
2
4x - x^{2}
4
x
−
x
2
.intersect
The radius in the Shell Method is always measured from the
axis of revolution
.
True
The limits of integration in the Shell Method are determined by the intersection points of the
bounding functions
.
True
Steps to determine the limits of integration in the Shell Method
1️⃣ Sketch the region bounded by the functions
2️⃣ Determine the points where the functions intersect
3️⃣ These intersection points define the limits of integration
What are the limits of integration if revolving the region bounded by f(x) = x^{2}</latex> and
g
(
x
)
=
g(x) =
g
(
x
)
=
4
x
4x
4
x
?
0 and 4