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===== The Cone: Instructor's Guide =====

==== Main Ideas ====

  * Visualization of 3-dimensional domains;
  * Use of cylindrical and spherical coordinates.

==== Students' Task ====
//Estimated Time: 30--45 minutes//

Students work in groups to write down and evaluate as many integrals as
possible for the volume of a cone.

==== Prerequisite Knowledge ====

  * Double integration in rectangular coordinates;
  * Basics of cylindrical and spherical coordinates;
  * Formulas for $dV$ in curvilinear coordinates.

==== Props/Equipment ====

  * [[:Props:start#whiteboards|Tabletop Whiteboard]] with markers
  * A handout for each student

==== Activity: Introduction ====

This activity encourages students to think of integration as chopping and
adding, emphasizing that there are many ways to chop.

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

Students may wonder whether to use single, double, and/or triple integrals.
Emphasize both that all of these choices are possible, and that the new
content in this activity involves the use of triple integrals.  Thus, if time
is short, emphasize the triple integrals.

Students may try to use constant limits; remind them that such limits
correspond to a box whose sides are described by holding some coordinate
constant.  Encourage students to draw pictures that show how they are
chopping, and to use those pictures to help determine the limits of
integration.

Students will likely realize that, for instance, in cylindrical coordinates
they will need to relate $z$ and $r$ along the edge of the cone, but will
likely have difficulty using proportional reasoning to simply write down the
answer.  Instead, they will likely try to use the general equation of a line,
and insert the appropriate slope and intercept.  Many students find this
construction surprisingly challenging, due both to the use of letters other
than $x$ and $y$, and to not being comfortable with the idea of the $(r,z)$
``plane''.

Most students will place the base of the cone in the $xy$-plane.  This choice
increases the computational difficulty in cylindrical coordinates, and makes
the computation in spherical coordinates quite challenging.  Encourage
students to consider putting the vertex of the cone at the origin, either by
turning it upside down, or by moving it below the $xy$-plane.

Getting the limits correct in spherical coordinates will be challenging for
most students.  The limits on $\theta$ aren't so bad (using $\tan\theta=R/H$),
but students will need to be shown that $z=r\,\cos\theta$, so that the top of
the cone is given by $r=H/\cos\theta$.

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

Emphasize that there are many ways to chop up the cone, including:
  * a single integral involving disks;
  * a single integral involving shells;
  * a double integral involving rings;
  * triple integrals in rectangular, cylindrical, or spherical coordinates;
and that the integrals in the latter two categories can be done in different
orders.

Have groups present one way of chopping up the cone, ideally including
triple integrals in cylindrical coordinates done in both orders.

==== Extensions ====

Introduce a density function for the density of chocolate inside the (solid)
cone, perhaps $\rho=3r^2$, and ask students to set up integrals for the total
amount of chocolate in the cone (or for the mass of the cone).  What are the
dimensions of ``$3$``?  (Grams per $\hbox{cm}^5$!)  What is $r$?  (Could be
spherical or cylindrical.)

A similar generalization is to fill the cone with ice cream, including a scoop
on top; the resulting ice cream cone is bounded by surfaces on which a
spherical coordinate is constant.