Volumes of Revolution: Cylindrical Shells

Learning Objectives

After studying this chapter, you should be able to:

• Calculate the volume of a solid of revolution by using the method of cylindrical shells.
• Compare the different methods (shells vs. disks/washers) for calculating a volume of revolution.
• Identify the correct variable of integration based on the chosen method and the axis of revolution.

Key Terms & Glossary

• Solid of Revolution: A three-dimensional figure formed by rotating a two-dimensional planar region around a straight line.
• Method of Cylindrical Shells: A technique for calculating volumes of solids of revolution by decomposing the solid into nested, hollow cylindrical tubes.
• Variable of Integration: The variable (usually $x$ or $y) with respect to which the integral is calculated.
• Axis of Revolution: The straight line (like the x-axis or y-axis) around which the 2D region is swept to create the solid.

The "Big Idea"

When calculating the volume of a solid of revolution, the Disk and Washer methods sometimes hit a mathematical wall. If your function is complex and defined as y = f(x), but you need to revolve it around the y-axis, the Washer method forces you to rewrite the function as x = g(y)$—which isn't always possible!

The Method of Cylindrical Shells circumvents this by slicing the region parallel to the axis of revolution, creating thin, nested cylinders (shells). Because of this geometry, we integrate along the coordinate axis perpendicular to the axis of revolution. This ability to choose your variable of integration based on the method makes Cylindrical Shells a powerful tool for complex functions.

Formula / Concept Box

[!IMPORTANT] The expression inside the integral for the shell method represents the surface area of a cylinder (2\pi r h) multiplied by the infinitesimally small thickness of the shell (dx$or$dy).

Hierarchical Outline

• 1. Introduction to Cylindrical Shells
  • Anatomy of a cylindrical shell (Radius, Height, Thickness)
  • Slicing parallel to the axis of revolution
• 2. The Shell Method vs. Disk/Washer Method
  • Disks/Washers: Integrate parallel to the axis of revolution
  • Shells: Integrate perpendicular to the axis of revolution
• 3. Formulating the Integral
  • Revolving around the y-axis (dx integration)
  • Revolving around the x-axis (dy integration)
• 4. Advanced Revolutions
  • Revolving around arbitrary vertical lines (x = k)
  • Revolving around arbitrary horizontal lines (y = k)

Visual Anchors

Diagram 1: Representative Rectangle for Shell Method

This diagram shows a representative vertical slice of thickness dx$ being rotated around the y-axis to form a cylinder.

Diagram 2: Method Selection Decision Tree

Use this logic to decide whether to use Shells or Disks/Washers based on your given equations.

Definition-Example Pairs

• Cylindrical Shell
  • Definition: A hollow 3D tube with an inner radius, an outer radius, and a height.
  • Real-World Example: A cardboard paper towel roll. The volume of the cardboard itself represents the volume of one shell.
• Integrating Perpendicular to the Axis
  • Definition: Using slices that step away from the axis of rotation at right angles.
  • Real-World Example: If revolving around a vertical flagpole (y-axis), you measure your steps horizontally along the ground ($dx) to map the shape.

Comparison Tables

Worked Examples

Example 1: Basic Volume using Cylindrical Shells

Problem: Define R as the region bounded above by the graph of y = 2x - x^2 and below by the x-axis. Find the volume of the solid generated by revolving R around the y-axis.

Step 1: Identify the integral components.

• We are revolving around the y-axis, using vertical slices (dx).
• Radius of the shell: r = x
• Height of the shell: h = f(x) = 2x - x^2
• Limits of integration: Find roots of 2x - x^2 = 0 \rightarrow x=0, x=2$.

Step 2: Set up the formula. $V = \int_{a}^{b} 2\pi x f(x) \, dx$ $V = \int_{0}^{2} 2\pi x (2x - x^2) \, dx$

Step 3: Evaluate the integral. $V = 2\pi \int_{0}^{2} (2x^2 - x^3) \, dx$ $V = 2\pi \left[ \frac{2x^3}{3} - \frac{x^4}{4} \right]_{0}^{2}$ $V = 2\pi \left( \left[ \frac{16}{3} - 4 \right] - 0 \right)$ $V = 2\pi \left( \frac{16}{3} - \frac{12}{3} \right) = 2\pi \left(\frac{4}{3}\right) = \frac{8\pi}{3}$

Example 2: The Volume of a Donut (Torus)

Problem: Find the volume of the torus created when the circle $x^2 + y^2 = r^2 is rotated around the vertical line x = R$ (where $R > r).

Step 1: Set up the geometry.

• Since we are rotating around a vertical line x = R, and the circle is a function of x, shells are perfect.
• Upper half of circle: y = \sqrt{r^2 - x^2}
• Lower half of circle: y = -\sqrt{r^2 - x^2}
• Total height of the shell: h = 2\sqrt{r^2 - x^2}

Step 2: Determine the radius.

• The distance from a point x inside the circle to the axis x = R$is$(R - x)$.

Step 3: Write the integral. $V = \int_{-r}^{r} 2\pi (R - x) \left( 2\sqrt{r^2 - x^2} \right) \, dx$

[!NOTE] Evaluating this requires splitting the integral and using trigonometric substitution or geometric formulas, but setting up the shell method properly is the hardest part!

Checkpoint Questions