Some triple integrals and parameterized surfaces
Find the volume (intersec) which is the between the curves
and
Solving these simultaneously gives us
or
The elliptical region
is the image of the union disk under the transformation
then
and we need only integrate
Find the center of mass of the volume (tunnels1 tunnels2) which is below the surfaces
and
and above the xy-plane.
Setting 
gives us the lines of intersection
y=x and y=-x. Since the surface we want is given
by
we see that the domain over which we integrate is
This can be broken up into four volumes with triangular bases, one of which is
Hence the center of mass is (0,0,a/b) where where
and
Find the volume (cylin1 cylin2 cylin3) which inside all three cylindars of the cylindars
and
Here the region in the xy plane to integrate over is the unit disk. Using symmetry we need only calculate the integral
which comes out to be 16/3.
Surface area can be calculated by adding up the areas of "tangent parallelograms". Here is a look at the tangent parallelograms as we successively make the grid size smaller: tangpar1 tangpar2 tangpar3 tangpar4
A spherical box looks like this:
sphbox1
sphbox2
sphbox3
It is approximately true that for small changes
in the spherical coordinates 
, 
, and 
,
that the volume of a spherical box is
Some students have commented that exam 2 had too many spherical coordinate problems on it and this constituted cruel and unusual punishment.
Other students enjoyed working on the extra credit problem of finding the volume of a cube using spherical coordinates. sphcube1 sphcube2 and they couldn't have asked for a better christmas present xcube1 xcube2 By the way the quickest way to "project" the unit sphere onto the cube
is:
The first three equation above give a surface parameterization of
the unit sphere1
sphere2. Here is an attempt
nsphere to
draw the parameterization space ( 
-plane) along
with the surface.
If we vary our spherical coordinates as follows:
We get the surface grapes.
The torus has the parameterization:
where a small circle of radius r is rotated about a large circle of radius R. The angle v determines the rotation about the z-axis. This same torus can be reparameterized by
To give us a twisted torus: twistor
Consider the parameterization:
What do you think is going to happen if we add (or subtract) a little bit to the z -coordinate? Answer
Another simple variation is to vary r as a function of u and-or v.
where r=sin(4 u) beads or where r=.5 u horn.
Two more surfaces which can be described by a Torus with a varing small circle r are the following: If we consider the polar cordinate equation
we get two circles of radius 1/2, one centered at (0,1/4) and the other centered at (0,-1/4). This is a figure eight. If we rotate the figure eight around a big circle we will get two toruses, one sitting above the other. If we also twist the figure eight as we go around we end up with two interlocking rings:
If we rotate the figure "eight" 1.5 times as we go around the big circle, we get a knot
A Mobius strip is a surface which is obtained by taking a rectangular strip of paper and twisting it. It has the surprising property that it has only one side. We can parameterize it by imagining we take a diagonal out of the small circle of a torus and twist it around.
Executable and source (part 3): progs3.zip. These programs are written in TrueBasic for MSDos computers and zipped into an archive using PKZIP (use PKUNZIP) to unzip.
