Tutorial 6: Volume of Solids of Revolution

Tutorial problems covering volume calculations using disk, washer, and shell methods.

Sections

Disk Method (Problems 1-4)

  • Revolution about x-axis
  • Revolution about y-axis
  • Single function revolution

Washer Method (Problems 5-8)

  • Region between two curves
  • Creating holes in solids
  • Choosing inner and outer radii

Shell Method & Mixed Problems (Problems 9-12)

  • Cylindrical shells
  • Choosing appropriate method
  • Complex regions

Method Selection Guide

Disk Method:

  • $V = \pi \int_a^b [f(x)]^2,dx$ (revolve about x-axis)
  • $V = \pi \int_c^d [g(y)]^2,dy$ (revolve about y-axis)

Washer Method:

  • $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2),dx$

Shell Method:

  • $V = 2\pi \int_a^b x \cdot f(x),dx$ (vertical shells, revolve about y-axis)
  • $V = 2\pi \int_c^d y \cdot g(y),dy$ (horizontal shells, revolve about x-axis)

Problem-Solving Approach

  1. Sketch the region
  2. Identify axis of revolution
  3. Choose method (disk/washer vs shell)
  4. Determine limits of integration
  5. Set up and evaluate integral

Links