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AP Calculus BC
Unit 8: Applications of Integration
8.13 Volume with Shell Method: Revolving Around the x- or y-Axis
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What is the shell method used to calculate?
Volume of solids
When revolving around the x-axis, the formula for the volume is
V = 2π∫_a^b y f(y) dy
The shell method is useful when
disk
or washer methods are difficult to set up.
The first step in setting up the shell method integral is to identify the axis of
revolution
The limits of integration in the shell method are determined based on the range of
x
x
x
or
y
y
y
.
What variable of integration is used when revolving around the x-axis in the shell method?
y
y
y
Arrange the key steps to set up the integral for the shell method in the correct order.
1️⃣ Identify the axis of revolution
2️⃣ Express functions in terms of the relevant variable
3️⃣ Determine the limits of integration
The shell method is particularly useful when disks or
washers
are difficult to set up.
What is the expression for
x
x
x
in terms of
y
y
y
for the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
?
x
=
x =
x
=
y
\sqrt{y}
y
To find the volume of the solid obtained by revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
about the y-axis, the limits of integration are from
0
0
0
to
2
2
2
.
The volume of the solid obtained by revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
about the y-axis is
8
π
8\pi
8
π
What is the formula for finding the volume when revolving around the y-axis using the shell method?
V
=
V =
V
=
2
π
∫
a
b
x
f
(
x
)
d
x
2\pi \int_{a}^{b} x f(x) \, dx
2
π
∫
a
b
x
f
(
x
)
d
x
When revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
about the y-axis, x</latex> in terms of
y
y
y
is \sqrt{y}
What is the integral setup to find the volume when revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
about the y-axis?
V
=
V =
V
=
2
π
∫
0
4
y
y
d
y
2\pi \int_{0}^{4} y \sqrt{y} \, dy
2
π
∫
0
4
y
y
d
y
Identifying the axis of revolution is essential for setting up the correct integral in the
shell method
.
Match the axis of revolution with its corresponding shell orientation and variable of integration:
x-axis ↔️ Vertical shells,
y
y
y
y-axis ↔️ Horizontal shells,
x
x
x
What is the formula for finding the volume when revolving around the x-axis using the shell method?
V
=
V =
V
=
2
π
∫
a
b
y
f
(
y
)
d
y
2\pi \int_{a}^{b} y f(y) \, dy
2
π
∫
a
b
y
f
(
y
)
d
y
When revolving the region bounded by
y
=
y =
y
=
x
2
x^{2}
x
2
and
y
=
y =
y
=
4
4
4
about the y-axis, the height of the shells is 4 - x^{2}
Revolving around the y-axis requires horizontal shells and integrating with respect to
x
x
x
.
Steps to set up the integral for volume using the shell method:
1️⃣ Identify the axis of revolution
2️⃣ Express functions in terms of the correct variable
3️⃣ Determine limits of integration
What is the formula for volume when revolving around the y-axis using the shell method?
V
=
V =
V
=
2
π
∫
a
b
x
f
(
x
)
d
x
2\pi \int_{a}^{b} x f(x) \, dx
2
π
∫
a
b
x
f
(
x
)
d
x
What is the formula for volume when revolving around the x-axis using the shell method?
V = 2\pi \int_{a}^{b} y f(y) \, dy</latex>
The axis of revolution determines the orientation of the cylindrical
shells
Steps to evaluate the integral to find the volume using the shell method:
1️⃣ Set up the integral
2️⃣ Simplify the integrand
3️⃣ Find the antiderivative
4️⃣ Evaluate the antiderivative at the limits of integration
5️⃣ Calculate the final volume
What is the antiderivative of
4
x
−
x
3
4x - x^{3}
4
x
−
x
3
?
2x^{2} - \frac{x^{4}}{4}</latex>
What is 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 the y-axis?
8
π
8\pi
8
π