Mathematics 263                                 Laboratory 8                                    November 10, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

Multiple Integration

In this lab we learn to use Mathematica to evaluate double integrals, and apply our knowledge to a practical problem.

Part 1:  A Review of Integration on Mathematica

As most of you recall, Mathematica knows the basic techniques of integration we learned in Calculus I and II.  For instance, Mathematica can compute the definite integral

when we issue the command

Integrate[ x^2 Sin[x], {x, 0, Pi}]

Write the value of the integral here:

Use the Plot command below to generate a graph of this function. Print out a copy and shade in the area we have computed.

Plot[x^2 Sin[x], {x, 0, Pi}]

If instead we wanted to compute the iterated integral

we would use the command

Integrate[2 y Sin[x], {x, 0, Pi}, {y, 0, x}]

Write the value of the integral here:

The command below plots the surface z = 2y sinx, but it does not indicate the region over which the integration is to be performed.  Print out a copy and shade in the portion of the xy-plane over which the integration is performed.

Plot3D[2 y Sin[x],{x,0,Pi},{y,0,Pi},
       AxesLabel->{"x", "y", "z"},
       ViewPoint->{1.080, 3.202, 0.177}]

Part 2: An Application: Achille's Shield

As an example of the utility of multiple integration, consider the following scenario:  Following graduation, you seek your fotune in archaeology.  You and two partners begin digging at the site of the ancient city of Troy and unearth a priceless treasure of antiquity: the golden shield of Achilles, hero of the Trojan War.  You and your partners can't decide on ownership of the solid gold, jewel-studded shield, so you decide that the only fair thing to do is saw the shield into three equally sized pieces.

Suppose the thickness of the shield is uniform, and it is made of the same material throughout (gold).  Then three pieces will be equal if the area of their outlines are equal.  Let's suppose the shield is elliptical in its outline; then we would like to divide it into three pieces roughly as shown below

The area of an ellipse

Our first step must be to find the area of the entire shield, so we can determine what each third will consist of.

Some of you may remember the formula from high school geometry:

Suppose the outline of the shield has the equation

where x and y are given in "hands," ancient Trojan units of measurement.  How long will the major and minor axes be?  To answer this question, we will plot the ellipse.  Solve the equation given for y as a function of x.  Write your answer below, and use it to complete the following command.

f[x_] := (* Your definition goes here *)

Plot[{f[x],-f[x]}, {x,-3,3}, AspectRatio->Automatic]

Use the Plot command above to plot the ellipse, examine your plot, and print out a copy for later in the lab.  Write the lengths of the major and minor axes (in hands) and the area of the shield's outline (in square hands) below:

Suppose you didn't remember the area formula from high school.  Set up an integral of the form

that would compute the area of this ellipse.  Insert your definitions in the command below, and use it to find the area.

Integrate[1, {x, xmin, xmax}, {y, ymin, ymax}]

Your answer should agree with the high school formula answer.  What is the area of the outline of each of the pieces the shield will be sawed into?  Set your answer equal to thirds by completing the command below.

thirds = (* Your answer goes here *)

Finding where to saw

We will attempt to find the coordinate a of the point shown in the plot below.

Define the function f (a) by replacing xmax,  ymin, and  ymax in the command below with the values you used above.

f[a_] := Integrate[1, {x, a, xmax}, {y, ymin, ymax}]

What is the physical significance of  f (a)?  Why do we want to solve the equation

f[a] == thirds

Solve the equation given above.  Give the value of a to three decimal places. On the printout of the ellipse you made above, sketch in the cuts you should make to saw the shield into three pieces of equal size.