Applications of volume integrals include calculating mass from mass density, and calculating charge from charge density.
An example was explored [Transparancies 2 and 3] in which the volume integrated over was in the shape of a box with a slanted roof. It was shown that the order of integration doesn't matter, as long as one treats the integration limits properly. However, some choices are more complicated than others. For example, when integrating over x first, the integral must be split up into two different regions. However, when integrating over z or y first, the integral can be done as one region.
In order to save time, these integrals were evaluated using Maple. Here are the Maple commands used, along with the corresponding output:
int(int(int(x+y*z,z=0..(1+x)),x=0..1),y=0..2);
4
int(int(int(x+y*z,y=0..2),z=0..(1+x)),x=0..1);
4
int(int(int(x+y*z,z=0..(1+x)),y=0..2),x=0..1);
4
int(int(int(x+y*z,x=0..1),y=0..2),z=0..1)
+ int(int(int(x+y*z,x=(z-1)..1),y=0..2),z=1..2);
4
These examples also illustrate how to integrate in Maple, which
is needed on HW #08,
sec. 4.8 #4.
Another example was explored [Transparancy 4], this one involving a quarter section of a parabolic cylinder with a slanted roof. In this case, it makes sense to do the z integral first, otherwise the slanted roof is going to necessitate the volume being split up into two seperate regions.
This integral was also evaluated in Maple. Here is the Maple command used, along with the answer:
int(int(int(1,z=0..(2+x+y)),y=0..(1-x^2)),x=0..1);
37
--
20
After this, we moved on to consider a qualitative introduction to
The Divergence Theorem and Stoke's Theorem. We
have now reached the apex of the course. These are the
fundamental theorems which the whole course has been leading up
to, and for which you've been learning the tools to understand!
When we move on to chapter 5, these theorems will be considered
in more detail. For the current purposes, a more qualitative
discussion of the Divergence Theorem [Transparancy 5], and
Stokes's Theorem [Transparancy 6], is given.
Because we previously defined the divergence in terms of the net outflux per unit volume, we can see that the volume integral of divergence cancels out everywhere except on the surface of the region, and yields the total flux coming out of the surface. This is the Divergence Theorem!
Furthermore, remembering that we previously defined curl in terms of the "swirl" per unit area, we can now see that this "swirl" is just the line integral of the vector field around an infinitesimal rectangle, and that the surface integral of the curl is going to be the sum of an infinite number of line integrals around infinitesimal rectangles. These line integrals will obviously cancel on any inner boundary, leaving just the line integral around the whole outer boundary of the surface. This is Stoke's Theorem!
(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 60 and 75 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 225 K, which takes a bit longer to download.)
