• Integration involves multiplying (and chopping and adding).

Estimated Time: 15–30 minutes

Students work in groups to write down and evaluate single integrals for finding the volume of a cylinder.

• Ability to interpret contour maps is helpful

• Tabletop Whiteboard with markers
• Surface models and contour maps
• A handout for each student

A good introduction to this activity is the first part of the Surfaces activity (“On your Mark” and “Get Set”, but not “Go”).

Some students will realize immediately how their surface (“town”) fits on the contour map (“county”), but others may require help. (The maps do have marks for the colored points, which the instructor can use to determine the proper placement, with or without sharing that fact with the students.)

It is important for students to realize that the density of lead is essentially constant in the park (but not in the town). Thus, they need to estimate the density there, then multiply by the area of the park.

Most students figure out on their own what density they need to meet the target amount of lead, then look for a location with that density.

A discussion of the relative sizes of the park, town, and county is helpful. Most students will realize that this activity is setting up the idea of a double integral, emphasizing the “multiply” step (and the importance of $dA$, which can be emphasized later).

A natural followup question is to ask how students would determine the total amount of lead in the town. (Don't ask them to actually compute this quantity.) A natural followup activity that does something similar is the Cake activity.