Lecture Notes: Volume Using the Shell Method

Introduction to Shell Method

• Focus on finding the volume when a region is rotated about an axis using the shell method.
• Picture Drawing: Essential to draw a rectangle parallel to the axis of rotation.

Shell Method for Rotation about the Y-Axis

• Setup:
  • Limits of integration: from A to B.
  • Radius: distance between the rectangle and axis of rotation (y-axis).
  • Height: height of the shell (rectangle), parallel to the y-axis.
• Formula:
  • Volume = (2\pi \int_{a}^{b} \text{radius(x)} \times \text{height(x)} , dx)
• For Two Curves:
  • Top curve (f(x)), bottom curve (g(x))
  • Height (h(x) = f(x) - g(x))_

Shell Method for Rotation about the X-Axis

• Setup:
  • Rectangle parallel to the x-axis.
  • Limits of integration: from C to D (y-values).
• Formula:
  • Volume = (2\pi \int_{c}^{d} \text{radius(y)} \times \text{height(y)} , dy)
• For Two Functions:
  • Right function (f(x)), left function (g(x))
  • Height (h = f(x) - g(x))_

Example Problem 1

• Curve: (y = \sqrt{x}), region bounded by (y = 0), (x = 4), rotated about y-axis.
• Solution:
  • Radius (r = x).
  • Height (h = \sqrt{x}) (convert (y) to terms of (x)).
  • Integral Setup: (2\pi \int_{0}^{4} x \cdot \sqrt{x} , dx).
  • Integration Process:
    • Combine exponents: (x^{3/2}).
    • Antiderivative: (\frac{2}{5} x^{5/2}).
    • Evaluate from 0 to 4: Result is (\frac{128\pi}{5})._

Example Problem 2

• Curve: (y = x - x^3), region bounded by (y = 0), (x = 0) to (x = 1), rotated about y-axis.
• Solution:
  • Find x-intercepts: 0, -1, +1 (focus on right side, 0 to 1).
  • Radius (r = x).
  • Height (h = x - x^3).
  • Integral Setup: (2\pi \int_{0}^{1} x(x - x^3) , dx).
  • Integration Process:
    • Distribute (x): (x^2 - x^4).
    • Antiderivatives: (\frac{x^3}{3} - \frac{x^5}{5}).
    • Evaluate from 0 to 1: Result is (\frac{4\pi}{15})._

Summary

• The shell method is useful for finding the volume of solids of revolution about an axis.
• Ensure the rectangle is parallel to the axis of rotation.
• Convert functions to terms of the variable corresponding to the axis of rotation.
• Utilize the formula (2\pi \int \text{radius} \times \text{height}) for calculations.