Section 5.3: Volumes by Cylindrical Shells

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Course

MATH162

Overview

The shell method provides an alternative to the disk/washer method for computing volumes of solids of revolution. While washers slice perpendicular to the axis of rotation, shells slice parallel to it—each thin slice becomes a cylindrical tube when rotated.

Why learn another method? Some problems that are nightmares with washers become straightforward with shells. If rotating about the $y$-axis requires solving a cubic for $x$, the shell method lets you avoid that entirely.

Key Equations

Formula Description When to Use
$V = 2\pi rh \Delta r$ Volume of a single shell Understanding the method
$V = \int_a^b 2\pi x f(x) \, dx$ Rotation about $y$-axis Region given as $y = f(x)$
$V = \int_c^d 2\pi y g(y) \, dy$ Rotation about $x$-axis Region given as $x = g(y)$
$V = \int_a^b 2\pi \lvert x - k \rvert f(x) \, dx$ Rotation about $x = k$ Non-standard vertical axis

The Universal Pattern: \(V = \int 2\pi \cdot (\text{radius}) \cdot (\text{height}) \cdot d(\text{thickness})\)

Skills in This Section

Skill Description Difficulty
Shell Method: Non-Standard Axes of Rotation Volumes of Revolution Intermediate
Shell Method: Rotation About the y-axis Volumes of Revolution Intermediate
Shells vs. Washers: Choosing the Right Method Volumes of Revolution Intermediate
The Shell Method Formula Volumes of Revolution Beginner

Learning Path

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    subgraph "From Section 5.2"
        A["Disk/Washer<br/>Method"]
    end

    subgraph "Section 5.3: Cylindrical Shells"
        B["Shell Method<br/>Formula"]
        C["Shell Method:<br/>y-axis"]
        D["Shell Method:<br/>Other Axes"]
        E["Shells vs<br/>Washers"]
    end

    subgraph "To Section 5.4"
        F["Work"]
    end

    A --> B
    B --> C
    C --> D
    D --> E
    E --> F

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    click B "../../skills/ch5-sec3/shell-method-formula.html"
    click C "../../skills/ch5-sec3/shell-method-y-axis.html"
    click D "../../skills/ch5-sec3/shell-method-other-axes.html"
    click E "../../skills/ch5-sec3/shells-vs-washers.html"
Phase Skill Key Takeaway
1 Shell Method Formula $V = 2\pi rh \Delta r$ = circumference × height × thickness
2 Shell Method: y-axis Vertical strips, integrate with $dx$
3 Shell Method: Other Axes Radius = distance to axis
4 Shells vs Washers Choose method based on which variable is easier

Exercise Coverage Map

Stewart Exercise Skill(s) Tested Difficulty
1-2 Shell setup, conceptual Level 1
3-4 Shell integral setup Level 2
5-8 Setup without evaluation Level 2
9-14 y-axis rotation Level 2-3
15-20 x-axis rotation with shells Level 3
21-22 Both variables Level 3
23-24 Non-standard axes Level 3
25-30 Rotation about $x = k$ or $y = k$ Level 3-4
31-36 Calculator/CAS problems Level 3-4
37-38 Midpoint Rule approximation Level 3
39-42 Describe the solid from integral Level 4
43-46 Technology-assisted Level 4
47-59 Mixed methods, choice of approach Level 4
60 Express $a$ in terms of $V$ Level 5
61-63 Derive formulas (sphere, etc.) Level 5
64 Napkin ring problem Level 5

Self-Assessment Quiz

Q1: What is the volume formula for a cylindrical shell?

$V = 2\pi rh \Delta r$

where $r$ is the average radius, $h$ is the height, and $\Delta r$ is the thickness.

Think of it as: circumference × height × thickness

Q2: When rotating about the y-axis, what is the radius of a shell at position x?

The radius is simply $x$ (the distance from the shell to the $y$-axis).

Q3: For rotation about $x = k$, what is the radius of a shell at position x?

The radius is $\vert x - k\vert $ (the distance from position $x$ to the axis $x = k$).

  • If $x > k$: radius = $x - k$
  • If $x < k$: radius = $k - x$

Q4: When should you use shells instead of washers?

Use shells when:

  • The function $y = f(x)$ is hard to solve for $x$
  • Rotating about a vertical axis with $y = f(x)$ given
  • The washer method would require splitting into multiple integrals

Use washers when:

  • The function is naturally given as $x = g(y)$
  • The inner/outer radii are easy to identify
  • Rotating about a horizontal axis with simple $y = f(x)$

Q5: What direction are the strips for shells vs washers?
  • Shells: Strips are parallel to the axis of rotation
  • Washers: Slices are perpendicular to the axis of rotation

For a vertical axis: shells use vertical strips ($dx$), washers use horizontal slices ($dy$).

Deep Connections

Why Does the Shell Method Work?

The shell method exploits a beautiful geometric fact: a thin cylindrical shell can be “unrolled” into a flat rectangular slab. The slab has:

So the volume is just length × height × thickness = $2\pi rh\Delta r$.

Connection to Pappus’s Theorem

The shell formula $V = 2\pi rh\Delta r$ is actually a special case of Pappus’s Centroid Theorem: the volume of a solid of revolution equals the area of the cross-section times the distance traveled by its centroid.

For a thin vertical strip rotated about the $y$-axis:

The Napkin Ring Surprise (Exercise 64)

One of the most surprising results in this section: if you drill a hole through the center of a sphere, the volume of the remaining “napkin ring” depends only on the height of the ring, not on the original sphere size or hole diameter!

Two napkin rings of the same height have the same volume, regardless of whether one came from a marble and one from a basketball.

Key Mathematical Themes

Theme How It Appears in Section 5.3
Alternative representations Same volume, two methods (shells vs washers)
Choosing the right tool Strategic decision based on problem structure
Distance as a fundamental quantity Radius = distance to axis
Decomposition into simpler shapes Solid → infinitely many thin shells
Riemann sums → integrals Sum of shell volumes → integral

Looking Ahead

The shell method completes your toolkit for volumes of revolution. In Section 5.4 (Work), you’ll see similar “slice and sum” reasoning applied to physics problems—computing work done by a variable force.

The pattern continues: identify what varies, set up the differential element, integrate.

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Section 5.2 Chapter 5 Section 5.4