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Calculating Volume with Shell Method

Mar 3, 2025

Lecture Notes: Volume Calculation Using the Shell Method

Introduction

  • Focus: Calculating volume using the shell method.
  • Objective: Find the volume of a solid of revolution, particularly when rotated about an axis.

Basic Concepts

  • Shell Method Formula: Volume = (2\pi \int_{a}^{b} \text{radius} \times \text{height} , dx)
  • Radius: Distance between the x-axis and the axis of rotation.
  • Height: Parallel to the axis of rotation._

Rotation About the y-axis

  • Rectangle Orientation: Parallel to the y-axis.
  • Radius and Height: Functions of (x).
  • Multiple Curves:
    • Top Curve: (f(x))
    • Bottom Curve: (g(x))
    • Height: (f(x) - g(x))
  • Limits of Integration: (x)-values, from (a) to (b).

Rotation About the x-axis

  • Rectangle Orientation: Parallel to the x-axis.
  • Radius and Height: Functions of (y).
  • Limits of Integration: (y)-values, from (c) to (d).
  • Multiple Functions:
    • Right Function: (f(x))
    • Left Function: (g(x))
    • Height: (f(x) - g(x))

Example 1: Rotating about the y-axis

  • Curve: (y = \sqrt{x}), bounded by (y = 0) and (x = 4).
  • Radius: (r = x) (distance from y-axis).
  • Height: (h = y = \sqrt{x}).
  • Volume Calculation:
    • Integrate (2\pi \int_{0}^{4} x \cdot \sqrt{x} , dx).
    • Simplify: (x \cdot x^{1/2} = x^{3/2}).
    • Integrate: (\frac{2}{5}x^{5/2}) from 0 to 4.
    • Result: (\frac{128\pi}{5})._

Example 2: Rotating about the y-axis

  • Curve: (y = x - x^3), bounded by (y = 0) and (x = 0) to (x = 1).
  • X-Intercepts: 0, -1, 1.
  • Radius: (r = x).
  • Height: (h = y = x - x^3).
  • Volume Calculation:
    • Integrate (2\pi \int_{0}^{1} x(x - x^3) , dx).
    • Simplify: (x^2 - x^4).
    • Integrate: (\frac{x^3}{3} - \frac{x^5}{5}) from 0 to 1.
    • Result: (\frac{4\pi}{15})._

Key Points

  • Always draw rectangles parallel to the axis of rotation.
  • Ensure that radius and height are expressed in terms of the variable of integration.
  • Understand how to adjust integration limits based on rotation axis.
  • Practice with multiple curves and understand how to calculate height.

These notes provide a framework for using the shell method in volume calculations and offer examples to reinforce understanding. Review the process of setting up integrals and practice problems to gain confidence in application.

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