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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 for in calculus?
Volume of solids
The formula for the shell method is
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
The radius in the shell method is the distance from the
axis of revolution
to the x-coordinate.
What are the limits of integration in the example when revolving
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
about
x
=
x =
x
=
3
3
3
?
−
2
- 2
−
2
and
2
2
2
The volume of the solid in the example is
64
π
64\pi
64
π
What is the formula for the radius when revolving around a vertical axis
x
=
x =
x
=
a
a
a
?
r
(
x
)
=
r(x) =
r
(
x
)
=
∣
x
−
a
∣
|x - a|
∣
x
−
a
∣
When revolving around a horizontal axis, the radius is
r(y)
= |y - b|</latex>
What is the height formula when revolving around a horizontal axis?
h
(
y
)
=
h(y) =
h
(
y
)
=
f
(
y
)
−
g
(
y
)
f(y) - g(y)
f
(
y
)
−
g
(
y
)
The formula for the shell method is
V = 2\pi \int_{a}^{b} r(x) h(x) dx</latex>
What is the height of the shell when revolving
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
around
x
=
x =
x
=
3
3
3
?
h
(
x
)
=
h(x) =
h
(
x
)
=
4
−
x
2
4 - x^{2}
4
−
x
2
What are the limits of integration for the integral in the example when revolving around the line
x
=
x =
x
=
3
3
3
?
x
=
x =
x
=
−
2
- 2
−
2
to
x
=
x =
x
=
2
2
2
When revolving a region around a line other than a coordinate axis, the
shell
method is used.
The formula for the shell method isV = 2\pi \int_{a}^{b} r(x) h(x)
dx
</latex>
What does
h
(
x
)
h(x)
h
(
x
)
represent in the shell method formula?
Height of the shell
An example region is bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
when revolved around the line
x
=
x =
x
=
3
3
3
, the radius is 3 - x.
What is the height
h
(
x
)
h(x)
h
(
x
)
in the example when revolving around
x
=
x =
x
=
3
3
3
?
4
−
x
2
4 - x^{2}
4
−
x
2
The volume of the solid when revolving around
x
=
x =
x
=
3
3
3
is
64
π
64\pi
64
π
cubic units.
What is the formula for the radius
r
(
x
)
r(x)
r
(
x
)
when revolving around a vertical axis of revolution
x
=
x =
x
=
a
a
a
?
r
(
x
)
=
r(x) =
r
(
x
)
=
∣
x
−
a
∣
|x - a|
∣
x
−
a
∣
When revolving around a horizontal axis of revolution
y
=
y =
y
=
b
b
b
, the radius is given by r(y) = |y - b|.
The height
h
(
y
)
h(y)
h
(
y
)
when revolving around
y
=
y =
y
=
−
2
- 2
−
2
is
4
−
y
2
4 - y^{2}
4
−
y
2
.
How is the radius defined when using the shell method to revolve around a vertical axis?
r
(
x
)
=
r(x) =
r
(
x
)
=
∣
x
−
a
∣
|x - a|
∣
x
−
a
∣
When revolving around a vertical axis, the integration variable is
x
The radius in the shell method is always positive regardless of the
axis of revolution
.
What are the limits of integration when revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
around
x
=
x =
x
=
3
3
3
?
x
=
x =
x
=
−
2
- 2
−
2
to
x
=
x =
x
=
2
2
2
The formula for the shell method with a horizontal axis is
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 axis of revolution is horizontal, we use
y
y
y
and
d
y
dy
d
y
as the integration variable.
How is the radius defined when revolving around a vertical line
x
=
x =
x
=
a
a
a
?
r
(
x
)
=
r(x) =
r
(
x
)
=
∣
x
−
a
∣
|x - a|
∣
x
−
a
∣
The height in the example is
4
−
x
2
4 - x^{2}
4
−
x
2
What is the integral for the volume when revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
around
x
=
x =
x
=
3
3
3
?
V
=
V =
V
=
2
π
∫
−
2
2
∣
x
−
3
∣
(
4
−
x
2
)
d
x
2\pi \int_{ - 2}^{2} |x - 3| (4 - x^{2}) dx
2
π
∫
−
2
2
∣
x
−
3∣
(
4
−
x
2
)
d
x
When revolving around a horizontal axis, the radius is r(y) =
|y - b|
</latex>.
The formula for the shell method with a vertical axis 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
What does the radius represent in the shell method?
Distance to the axis
Steps to set up the integral for volume using the shell method
1️⃣ Determine the axis of revolution
2️⃣ Define the radius
r
(
x
)
r(x)
r
(
x
)
or
r
(
y
)
r(y)
r
(
y
)
3️⃣ Identify the height
h
(
x
)
h(x)
h
(
x
)
or
h
(
y
)
h(y)
h
(
y
)
4️⃣ Find the limits of integration
The first step to evaluate an integral set up using the shell method is to expand the
integrand
.
After expanding the integrand, the next step is to find its
antiderivative
What is the final step in evaluating the integral in the shell method?
Substitute the limits
The volume of the solid formed by revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
around
x
=
x =
x
=
3
3
3
is
64
π
64\pi
64
π
cubic units.
When revolving around
x
=
x =
x
=
3
3
3
, the radius is
3
−
x
3 - x
3
−
x
Match the component with its description in the shell method:
Radius ↔️ Distance from the axis
Height ↔️ Bounding functions
Limits ↔️ Intersection points
Formula ↔️
2
π
∫
r
h
d
x
2\pi \int r h dx
2
π
∫
r
h
d
x
The function to integrate in the shell method is
r(x) h(x)
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