Lecture Notes: Volume Using the Shell Method

Introduction

• Focus: Finding volume using the shell method
• Example: Rotating about the y-axis

Basic Concepts

• Rectangle Orientation: Always draw the rectangle parallel to the axis of rotation (y-axis in this case).
• Radius: Distance between the x-axis and the axis of rotation.
• Height: The height of the shell or rectangle.
• Formula:
  • Volume = 2π ∫[a to b] (radius in terms of x) * (height in terms of x) dx*

Single Curve

• Limits of integration: from A to B (in terms of x).
• Rectangle parallel to the y-axis.
• Example: Shell height.

Multiple Curves

• If there are two curves, denote them as f (top) and g (bottom).
• Height of the shell: h(x) = f(x) - g(x).
• Rectangle still parallel to the y-axis.

Rotating About the X-Axis

• Rectangle should be drawn parallel to the x-axis.
• Radius: Distance between the rectangle and the axis of rotation.
• Height: Parallel to the axis of rotation.
• Formula:
  • Volume = 2π ∫[c to d] (radius in terms of y) * (height in terms of y) dy*

Single Curve

• Limits of integration: from C to D (in terms of y).
• Rectangle parallel to the x-axis.
• Example: Shell height.

Multiple Curves

• If there are two functions, denote f (right) and g (left).
• Height of the shell: h(x) = f(x) - g(x).
• Rectangle still parallel to the x-axis.

Example 1: y = sqrt(x) Rotated about the Y-Axis

• Curve: y = sqrt(x)
• Region: Bounded by y=0, x=4
• Rotate about y-axis
• Draw rectangle: Radius r = x, Height h = y
• Steps:
  1. Express radius and height in terms of x.
  2. Given y = sqrt(x), replace y in terms of x.
  3. Volume formula: 2π ∫[0 to 4] x * sqrt(x) dx
  4. Simplify: x * x^(1/2) = x^(3/2)
  5. Integrate: (2/5) x^(5/2) | from 0 to 4
  6. Compute: 2π (2/5 * 4^(5/2))
  7. Simplify: 4^(1/2) = 2; 2^5 = 32; 2π * (232/5) = 128π/5

Example 2: y = x - x^3 Rotated about the Y-Axis

• Curve: y = x - x^3
• Region: Bounded by y=0, from x=0 to x=1
• Rotate about y-axis
• Draw rectangle: Radius r = x, Height h = y
• Steps:
  1. Express radius and height in terms of x.
  2. Given y = x - x^3, replace y in terms of x.
  3. Volume formula: 2π ∫[0 to 1] x (x - x^3) dx
  4. Simplify: x(x - x^3) = x^2 - x^4
  5. Integrate: (1/3)x^3 - (1/5)x^5 | from 0 to 1
  6. Compute: 2π (1/3 - 1/5) = 2π (5/15 - 3/15)
  7. Simplify: 2π * (2/15) = 4π/15*