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📚 Learn How to Find Volumes Using the Cylindrical Shell Method
In this video, you’ll learn how to use the cylindrical shells method to find the volume of a solid of revolution. We’ll walk through the main formulas, explain how to identify the radius and height of each shell, and then apply the method to four complete examples involving rotation about vertical and horizontal lines.

🔍 What You’ll Learn in This Video
✅ The Cylindrical Shells Method
✔ A cylindrical shell has volume:
  V_shell = 2π(r)(h)(thickness)
✔ When rotating a region about a vertical line, integrate with respect to x
✔ When rotating about a horizontal line, integrate with respect to y
✔ Identify:
 • r = distance from the shell to the axis of rotation
 • h = height of the shell (difference of functions)

✅ Examples Covered in This Video
Example 1 — Rotating a Region About the y-axis
Region bounded by:
• y = 2x² − x³
• y = 0
Rotated about the y-axis, using r = x and h = 2x² − x³.

Example 2 — Rotating a Region About the Line x = a
Region bounded by:
• y = x − x²
• y = 0
Rotated about the vertical line x = 2, using r = 2 − x and h = x − x².

Example 3 — Rotating a Region About the x-axis
Region bounded by:
• x = 1 + y²
• x = 0
• y = 1
• y = 2
Rotated about the x-axis, using r = y and h = (1 + y²).

Example 4 — Rotating a Region About the Line y = b
Region bounded by:
• x = 1 + y²
• x = 5
• y = 0
Rotated about the horizontal line y = 3, using r = 3 − y and h = 5 − (1 + y²).

📌 Summary of Steps

1️⃣ Graph the curves
2️⃣ Find points of intersection
3️⃣ Determine the radius or radii from the graph
4️⃣ Choose the correct cylindrical shell formula
5️⃣ Evaluate the definite integral

By the end of this video, you’ll understand how to use cylindrical shells to find volumes of solids of revolution and how to set up integrals correctly for each type of rotation.

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Timestamps
00:00 Introduction
00:33 Formulas and Steps to Follow
07:34 Example - Revolving a region about the y-axis
12:50 Example - Revolving a region about the line x=a
17:52 Example - Revolving a region about the x-axis
22:55 Example - Revolving a region about the line y = b

More About Understand The Math
Understand The Math offers clear, accessible, and engaging mathematics instruction for students and lifelong learners. I'm Cheryl Hile, the founder of Understand The Math, with a Ph.D. in Engineering Science and Applied Mathematics and over 25 years of university teaching experience. My goal is to provide accurate, high-quality math instruction through step-by-step explanations and carefully worked-out examples.

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