One Cool Volume

Notes about this lab:  Steps in the solution of a volume/slicing problem are offset with 
    1) Bold Text.  And explanations.
Instructions for Mathematica commands are shown with
         Indented, italic text.
Specific problems requiring answers are posed in 
    Problems: Which are offset from other text.
You should respond to these by typing your solutions in the notebook in nicely formatted text.

Volumes and Arclength

(From our book comes the following simply splendid problem:)
Find the volume enclosed by two cylinders, with radii 2, lying respectivly along the x  and y  axes.

1) Visualize these.  Can you see what the shape looks like?  We can visualize it with Mathematica by using the SurfaceOfRevolution command, as follows.

First, load the Graphics`SurfaceOfRevolution` package by using the Needs["■`□`"] command (on the LocalMath palette—you will need to fill in the boxes in the following):

Needs["\[Placeholder]`\[Placeholder]`"]

This lets us use the SurfaceOfRevolution command, which has several different options.  The next command plots the first of the cylinders; we'll explain it after you run it to see what it looks like.

cyl1  =  
      SurfaceOfRevolution[2, {x, -2, 2}, RevolutionAxis -> {1, 0, 0}, 
        AxesLabel -> {"x", "y", "z"}, PlotPoints -> 45];

The "2" is the graph of a function in the x-z plane, the x-range gives the x values to plot, and the revolution axis is a point at the other end of a line from the origin around which to revolve the graph.  To get the second cylinder, we don't want a function in the x-z plane, and so use a different form of the command:

cyl2 = 
      SurfaceOfRevolution[{0, t, 2}, {t, -2, 2}, 
        RevolutionAxis -> {0, 1, 0}, PlotPoints -> 45];

Problem 1: Explain what this command is doing (that is, what is it graphing, and how is the axis of revolution defined?).

Fill in the following to show the two cylinders together

Show[\[Square]];

It would be nice to add ends to the cylinders.  The following surface of revolution command is an end for one of the cylinders:

end1 = 
      SurfaceOfRevolution[{2, 0, t}, {t, 0, 2}, RevolutionAxis -> {1, 0, 0}];

Problem 2: Write and execute a SurfaceOfRevolution command to graph the end of the other cylinder by finishing the following line:

end2 = 
      SurfaceOfRevolution[{\[Square], \[Square], \[Square]}, {
          \[Square], \[Square], \[Square]}, 
        RevolutionAxis -> {\[Square], \[Square], \[Square]}];

After doing problem 2, show your two ends together with the two cylinders below.

Show[\[Square]];

Do you get what you expect?  Which end of the y axis is at the front of the graph you see?  We can turn the graph around by using the ViewPoint option—there is a 3D ViewPoint Selector under "Input" on the command bar at the top of the screen.  Ask your instructor how to use this if you do not already.

Finally, add the other two ends of the cylinders—make a graph for each by filling in an appropriate SurfaceOfRevolution command for each of

\(end3 = 
\(end4 = \)\)

and show it together with the cylinders and the first two ends:

fullpic  =  Show[\[Square]];

1A) Visualize the slices.  How should you slice this volume?  The answer to this question is the
answer to

Problem 3: How should you slice shape to calculate the volume (i.e., vertically or horizontally)?  Why?  Explain.

Now that we've answered the question, let's try and visualize these slices with Mathematica.  We'll do this by plotting a plane to slice the shape and showing it with the full picture you created before.  Run the following command to generate the plane.

slicingplane = Plot3D[1.65, {x, -2, 2}, {y, -2, 2}];

Then show it with your fullpic from above.

Show[\[Square]];

You might want to reset the ViewPoint, and it might also be useful to reset the PlotRange to something like {-2,1.7}.

2) Work out the Volume of a Slice.  Next, find the volume of a typical slice, starting with the following problems.

Problem 4: On a separate sheet of paper, sketch a slice of the cylinders.  What dimension(s) do we know?  Which do we need to find?

Problem 5: What will the integration variable in your volume integral be?  What variable should the volume of each of your slices depend on?

Problem 6: What is the equation for the circle at one of the ends of one of the cylinders?  Why might this be useful?

The following plots a graph that may be useful in your finding the volume of a slice.

\(p1 = 
      Plot[\@\(4 - x\^2\), {x, \(-2\), 2}, AspectRatio -> 0.5, 
        DisplayFunction -> Identity]; 

    p2 = Plot[1.65, {x, \(-1.1302654555\), 1.1302654555}, 
        DisplayFunction -> Identity]; 

    p3 = Plot[1.45983405217\ x, {x, 0, 1.1302654555}, 
        DisplayFunction -> Identity]; 

    p4 = Plot[\(-1.45983405217\)\ x, {x, \(-1.1302654555\), 0}, 
        DisplayFunction -> Identity]; 

    Show[p1, p2, p3, p4, DisplayFunction -> $DisplayFunction]; \)

Problem 7: Find the volume of one slice of the two cylinders.

3) Integrate to Find the Volume.

Problem 8: Find the volume of the shape.  You may use Mathematica to do the integral for you.

Probability and such fun stuff

Read the Focus on Modeling section at the end of chapter 8.  Then use Mathematica to find the following.

The following defines a normal distribution with mean 100 and standard deviation 15.

normdist[x_] := 
      (1/(15 (Sqrt[ 2*π]))) 
        (E^(-(((x - 100))^2)/(2*(15^2))));

Problem 9: Suppose that this gives the probability that a person will have an IQ x.  Write an integral that gives the probability that a person will have an IQ between 115 and 120.  Use Mathematica to evaluate this.

If the answer you get doesn't prove too edifying, you might want to use the "numerical operator" N[□] to change it to a decimal answer.

Problem 10: If the height of adult American men is normally distributed with mean 69 inches and standard deviation 2.5 inches, find the percentage of men with heights between 66.5 and 71.5 inches.  Then find the percentage between 64 and 74 inches.  How are these ranges related to the mean and standard deviation of the probability distribution?