Today we are going to work on a mathematical problem in groups. Working effectively with a group is an important communication skill to develop. Whether you end up working in business, industry, or academe, you will need to explain your ideas to groups of people and to get your points across in meetings. One of the main skills that employers look for in prospective employees is the ability to work in groups.
Do you still remember your group from the first day of class? I would like you to work today with your original group, even if you are used to working with someone else in the class. This is good practice for the real world, where you will frequently have to work with people you do not know well.
Groups function better if each member has a defined role. Here are some of the different functions group members can perform.
The Secretary keeps a record of the major points of discussion.
The Gatekeeper ensures that every member's ideas are heard by encouraging silent members to speak and by holding back dominant members.
The Time Keeper keeps the group moving so that the task is completed on time.
The Devil's Advocate questions the consensus of the group to ensure that all sides of an issue are considered.
The Encourager gives positive reinforcement to other members.
Here is the plan for today's class:
10 minutes: organize groups and formulate a problem-solving strategy.
60 minutes: solve the problem.
20 minutes: prepare a presentation.
10 minutes per group: give the presentation.
Solving this week's homework depends on understanding today's problems, so it is important for your group to give a presentation that the other students can follow. Another group will be working independently on the same problem as your group, so we will have an opportunity to compare different approaches.
A classic exercise in third-semester calculus is to find the volume of the intersection of two cylinders. Consider two cylinders of unit radius, one cylinder having its axis along the x-axis, and the second cylinder having its axis along the y-axis. The problem is to find the volume of the intersection, that is, the volume of the region that is inside both cylinders simultaneously.
The surprise is that a square appears in the solution of this problem!
Your task is to use Maple to create graphics to help visualize the problem and then to solve the problem. You may use Maple to compute the integrals that arise.
Hint: The Maple plot option style=contour may be
useful.
From elementary plane geometry, you know that a circle of radius r has circumference 2*Pi*r and area Pi*r2. In three-dimensional space, a sphere of radius r has surface area 4*Pi*r2 and volume 4*Pi*r3/3.
Your task is to generalize to higher dimensions. What are values of the surface area and the volume of a ball of radius r in n-dimensional Euclidean space?
(This is a standard problem that you will find in many books. See, for example, James Stewart's book Calculus, third edition, Brooks/Cole, 1995, page 966, problem 12.)
You may use Maple to help with the computations.
Hints:
The Gamma function should appear in your answer.
(The Maple syntax is GAMMA(k), which equals
factorial(k-1) when k is a positive integer.)
There is a sneaky way to find the surface area. Namely, compute the integral over the whole space of exp(-x12-...-xn2) in two different ways: first as an n-fold rectangular integral, and second as an integral in generalized polar coordinates.
Created Oct 13, 1996.
Last modified Oct 16, 1996
by boas@tamu.edu.
URL: /~harold.boas/courses/696-96c/class7/activities.html
Copyright © 1996 by Harold P. Boas.
All rights reserved.