Overview
In this section, we begin to study some applications of Calculus to physics, engineering, and other sciences. Throughout Chapter 6, every application uses the same basic idea: “Slice” an object into small pieces, calculate something for each of those pieces, and then use an integral to add them all up. The idea is that we can calculate something when we slice an object into tiny pieces that we can’t calculate for the entire object all at once (think of the volumes and areas that we just studied – by slicing into rectangles and cylinders, we found simpler shapes).
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

State the definitions of mass, density, and center of mass.

State the formula relating mass, density, and volume.

State the formula for the center of mass of a collection of masses distributed along a single axis.

State the units of a definite integral, when given the units of the integrand and its variable.

State the formula for the mass of a rod whose mass is distributed with density function \(\rho(x)\) between \(x = a\) and \(x = b\).
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Use Calculus to calculate the mass of a thin rod or wire with variable density.

Use Calculus to find the center of mass of a thin rod or wire with variable density.

State clearly how the process of “slicing” and integrating allows us to do the above. In particular, explain why we can calculate the mass of a thin slice using the basic formula from physics, but why we cannot use the same formula for an entire object.
To prepare for class

Do the Preview Activity for this section (on WeBWorK if required by your teacher).

Read all of Section 6.3 in Active Calculus.

Watch the following videos (by GVSUmath):