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===== The Park: Instructor's Guide =====

==== Main Ideas ====

  * Integration involves multiplying (and chopping and adding).

==== Students' Task ====
//Estimated Time: 15--30 minutes//

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

==== Prerequisite Knowledge ====

  * Ability to interpret contour maps is helpful

==== Props/Equipment ====

  * [[:Props:start#whiteboards|Tabletop Whiteboard]] with markers
  * Surface models and contour maps **FIXME**
  * A handout for each student


==== Activity: Introduction ====

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

==== Activity: Student Conversations ====

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.

==== Activity: Wrap-up ====

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).

==== Extensions ====

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
[[courses:activities:mvact:mvcake|Cake]] activity.