Overview
Continuing our study of applications of integration, we will now look at some of the most powerful and useful ways to apply Calculus. We will see how to apply simple formulas to complex situations involving Work, Force, and Pressure. As in the rest of this chapter, “slicing” will play a key role, as we seek exact answers.
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 basic formula and units for Work.

Use the basic formula for Work when both force and distance are constant.

Explain in words why we cannot apply the standard formula for Work to situations such as lifting a leaky bucket and pumping water from a tank.

State and use an integral formula that lets you calculate Work in these situations.
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 work required in situations for which the force or distance varies.

Draw and clearly label a diagram that shows all relevant parts of a situation in which work is calculated.

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

Use Calculus to calculate the total hydrostatic force on an object.
To prepare for class
Use these resources to become proficient with the basic objectives (see above) before class.

Read Section 6.4 in Active Calculus up until (but excluding) subsection 6.4.3 “Force due to hydrostatic pressure”. (Read the remainder after class.)

Watch the following videos:

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