8.5 Physical Applications (Density, Mass, Work)

In this lesson, you will learn about some physical applications (density, mass, and work) of integration.

Lesson Objectives

Upon completion of the lesson 8.5, you will be able to:

• • Use integration to calculate the work done by a variable force.

  • Use integration to calculate the mass of an object with variable density.

  • Use integration to find the center of mass of a Lamina.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube videos: Work and Hooke's Law

  2. Example 1:

• 1. Example 2:

• 1. YouTube videos: Finding Work Using Calculus - The Cable/Rope Problem

  2. Part 1:

• 1. Part 2:

• 1. YouTube video: Calculating the Work Required to Drain a Tank

• 1. YouTube videos: Moments and Center of Mass

  2. Part 1 - Point Masses on a Line:

• 1. Part 2 - Point Masses in a Plane:

• 1. Part 3 - Planar Lamina of Uniform Density:

• 1. Part 4 - Integral Formula for Planar Lamina:

• 1. Part 5 - Rod of Non-uniform Density:

• 1. YouTube videos: Moments and Center of Mass

  2. Part 1:

• 1. Part 2:

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. Applications of Integration: Work

  2. Work

  3. Moments and Center of Mass

  4. Note: You need Flash or Java to be able to see the discussion.

  5. Center of Mass

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Work by a Variable Force

  2. Note: Read the section and do examples 1 - 3.

  3. Work

  4. Note: You need Flash or Java to be able to see the discussion or solutions to the problems.

  5. Work

  6. Centroid

  7. Note: Skip the parts of any problem that asks you to use Pappus' Theorem on Volume to calculate a volume. Just do the centroid part of the problems.

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. YouTube video: Work Done by a Variable Force

• 1. Section 6.4: Work

1. Work, Fluid Pressures and Forces

   1. Note: Skip the section on Fluid Pressure and Forces.

   2. YouTube video: Center of Mass, 1

• 1. YouTube videos: Centroid/Center of Mass

  2. Part 1:

• 1. Part 2: