The document describes the shell method for calculating the volume of solids of revolution. The shell method is used when the region is rotated about an axis that is parallel to the strips that make up the solid. Using this method, the volume of a cylindrical shell is calculated as V = 2πrth, where r is the average radius, t is the thickness, and h is the height. Several examples demonstrate how to set up and calculate the volume of solids rotated about different axes using the shell method formula.

2. SOLIDS OF
REVOLUTI
ON
Shell Method

3. The method is used when the element (or representative
strip) is parallel to the axis of revolution. When this strip is
revolved about the axis, the solid formed is of cylindrical form.
C. SHELL METHOD: V = 2πrth
3
If we slice perpendicular to
the y-axis, we get a washer.
But to compute the inner radius and the
outer radius of the washer, we’d have to
solve the cubic equation y = 2x2 – x3 for x in
terms of y; that’s not easy.
Fortunately, there is a method,
called the method of
cylindrical shells, that is
easier to use in such a case.
Figure in the right side shows a cylindrical shell
with inner radius r1, outer radius r2, and height
h.

4. Its volume V is calculated by subtracting the volume V1 of the
inner cylinder from the volume V2 of the outer cylinder:
C. SHELL METHOD: V = 2πrth
4
𝑉 = 𝑉2 − 𝑉1 = 𝜋𝑟2
2ℎ − 𝜋𝑟1
2ℎ
𝑉 = 𝜋 𝑟2
2
− 𝑟1
2
ℎ
𝑉 = 𝜋(𝑟2 + 𝑟1)(𝑟2 − 𝑟1)ℎ
𝑉 = 2𝜋
𝑟2 + 𝑟1
2
ℎ(𝑟2 − 𝑟1)
If we let 𝑟 = 𝑟2 – 𝑟1 (the thickness of the
shell) and 𝑟 =
1
2
(𝑟2 + 𝑟1) (the average
radius of the shell), then this formula for the
volume of a cylindrical shell becomes
and it can be remembered as
𝑽 = [𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆] [𝒉𝒆𝒊𝒈𝒉𝒕] [𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔]

5. C. SHELL METHOD: V = 2πrth
5
Now let S be the solid obtained by rotating about the y-axis the region
bounded by 𝑦 = 𝑓(𝑥) [where 𝑓 (𝑥)  0], 𝑦 = 0, 𝑥 = 𝑎, and 𝑥 = 𝑏, where
𝑏 > 𝑎  0.
But, from the definition of an integral, we know that

6. C. SHELL METHOD: V = 2πrth
6
The best way to remember formula is to think of a typical shell, cut and
flattened as in the figure below, with radius 𝑥, circumference 2𝑥, height
𝑓(𝑥), and thickness 𝑥 or 𝑑𝑥:
This type of reasoning will be helpful in other situations, such as when we
rotate about lines other than the y-axis.

7. Example 8
7
Find the volume of the solid generated by revolving the second quadrant
region bounded by the curve 𝑥2 = 4 − 𝑦 about 𝑥 − 1 = 0.
𝑥2
= 4 − 𝑦
𝑥 = 1 (𝐴𝑂𝑅)

8. Example 8
8
Find the volume of the solid generated by revolving the second quadrant
region bounded by the curve 𝑥2 = 4 − 𝑦 about 𝑥 − 1 = 0.

9. Example 9
9
The region bounded by the curve 𝑦 = 𝑥, the 𝑥–axis and the line 𝑥 = 4 is
revolved about the 𝑦-axis to generate a solid. Find the volume of the solid.
𝑉 =

10. Example 9
10
The region bounded by the curve 𝑦 = 𝑥, the 𝑥–axis and the line 𝑥 = 4 is
revolved about the 𝑦-axis to generate a solid. Find the volume of the solid.

11. Example 9
11
Determine the volume of the solid obtained by rotating the region bounded
by y = 2x+1, y = 3 and x = 4 about the line y = 10
𝑦 = 2𝑥 + 1 𝑥 = 4
𝑦 = 3
𝑦 = 10 (𝐴𝑂𝑅)
𝑉 = 90𝜋 𝑢3
A. Sketch B. AOR : y = 10
C. Boundaries: 𝑦1 = 3
4 =
𝑦−1
2
𝑦2 = 9
D. Radius: r = 10 - y
E. Set up: V = 2π 3
9
{(10 − 𝑦)[4 − (
𝑦−1
2
)]}𝑑𝑦
= 2π 3
9
(40 − 5𝑦 + 5 − 4𝑦 +
𝑦2−𝑦
2
)dy
F. Integrate:
V = 2𝜋[40y -
5𝑦2
2
+ 5y − 2𝑦2 + (
1
2
)(
𝑦3
3
−
𝑦2
2
)]3
9

12. Example 10
12
Determine the volume of the solid obtained by rotating the region bounded by x = 𝑦2,
y = 1 and x = 4 about the x-axis
𝑥 = 𝑦2
y = 1
𝑦 = 0
x= 4
𝑉 =
9𝜋
2
𝑢3
B. AOR : y = 0
C. Boundaries: 𝑦1 = 1
4 = 𝑦2
𝑦2 = 2
D. Radius: r = y
E. Set up: V = 2π 1
2
[(𝑦)(4 − 𝑦2)]𝑑𝑦
= 2π 1
2
(4𝑦 − 𝑦3
)dy
Integrate:
V = 2𝜋[2𝑦2
+
𝑦4
4
]1
2
A. Sketch