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The Cone: Instructor's Guide

Main Ideas

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

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.

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.

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