Talk:Calculus/Volume

From Wikibooks, the open-content textbooks collection

Contents 1 HTML vs. TeX formatting 2 Defining Volume as Cross-sectional Integral 3 Rigor and Target Audience 4 Possible Material 5 Possibly an error in sphere volume equation?

[edit] HTML vs. TeX formatting

In my recent edit to the first paragraph I changed some of the formula's that were not displayed equations to HTML formatting, because this makes line spacing cleaner. I understand this can be kind of a controversial change. What do you think? I am not particularly opposed to going back, but it might be nice to discuss.

I don't have a strong opinion on it either, except that I'm better at TeX than HTML (although neither is terribly difficult). I understand the line spacing issue, but I do think the TeX still looks nicer. Obviously, we should be consistent, though. W3asal (talk) 00:47, 30 January 2009 (UTC)

Ok, after some thought let go with latex'ed formula's. It will probably bother calculus students if the font their variables are written in keeps changing, and this is probably more important the typographical concerns. Besides I think that most of the rest of the book uses this convention. Now I don't know if it would help to have some strategy going into this section. Is the goal to mainly talk about surfaces of revolution and problems like that?

[edit] Defining Volume as Cross-sectional Integral

The way the section is currently written it seems like we are defining volume to be the integral of the cross-sectional area, which I think of more as a theorem.

I think I see your point here, and the wording of the section is probably a bit strong in terms of the fundamental nature of what volume is (it's certainly not intuitive to think of volume as the integral of cross-sectional area). I'll have to put some more thought into a better wording, but you should feel free to change it up however you see fit (I certainly won't take offense). W3asal (talk) 02:29, 31 January 2009 (UTC)

I have made some attempt. I find the sentence I wrote "Generally, if S is a solid that lies in between x = a and x = b, let A(x) denote the area of a cross section taken in the plane perpendicular to the x direction, and passing through the point x." awkward and it should be reworked. One particular problem I have is if the formula is meant is the over use of x. Also, do we mean in this paragraph that we are talking specifically about the x in the x, y, z plane. Of course we could make a point that we could work in any direction, but if we state the definition in terms of this then it becomes very tempting to start saying words like vector, which probably should be avoided. Probably we should just say something along the lines that, "the volume of an object doesn't change as you rotate it, so for simplicity of notation we state our definition as if we are taking our cross sections in the y-z plane, and the height in the x direction". But I am not sure if we should assume comfort with x, y, z coordinates if this is the first time they encountered volume. Anyways those are my random thoughts for today. Thenub314 (talk) 09:20, 1 February 2009 (UTC)

[edit] Rigor and Target Audience

I am not sure how much rigor we are aiming for, should this book be used for a course to math majors, a course for general science majors, or a course for very introductory high school students. I am most comfortable writing for math majors but I feel like we should aim for the more general group of science majors, but it seems like maybe the book is aimed at high school students. I have started looking at some sources to see what they say and what we should include.

I certainly don't think we should be writing for math majors (they need the rigor of an intro to analysis course). In my mind, I'm writing for math-minded high schoolers and average science majors. W3asal (talk) 02:29, 31 January 2009 (UTC)

[edit] Possible Material

Here are some thoughts about things we may want to indclude (probably a very incomplete list.)

• Surfaces of revolution including "Shell", "washer" and "disc" methods.
• Papuss's theorem about the volume of a region that is rotated being related to the distance the centroid moves. (Q: Have centroid's come up yet?)
• Other examples were we just are given how the cross sectional area behaves. (We are touching on this a bit now, with circles, cylinders etc) but maybe volumes of general cones.
• Examples of shapes with finite volume but infinite surface area.

Does anyone know how to make some nice graphics for this material (which lends itself quite well to graphical representation)? W3asal (talk) 02:29, 31 January 2009 (UTC)

Well I know enough that I could figure out how to do something with a lot of work. In general we shouldn't underestimate the possibility of transwiki'ing some images from the articles on wikipedia, but I don't think it helps with this particular instance. Thenub314 (talk) 10:58, 31 January 2009 (UTC)

Which of these files do you think is better? I personally like no box, but takes some editing with gimp. These are examples of things I might accomplish, but I could use some help deciding which images would best. Thenub314 (talk) 20:46, 1 February 2009 (UTC)

I agree that no box looks better, but the box is not a big deal. What software did you use to make them? W3asal (talk) 22:35, 2 February 2009 (UTC)

I used on open source mathematics program called Sage. As I understand it is a front end to a lot of other open source packages. I hadn't used it before but they had a page with example pictures. Starting with the example of a cylinder inside of a cone, I played around until I got these (as I said I just used gimp to take out the box). I don't know if there is a way to put nice latex labels in, maybe just editing them in after? Probably this would be easier in mathematica or maple, but I don't have access to those at the moment. It should be fairly easy to do surfaces of revolution using this method as well. Drawing a generic blob in space might be harder (unless you know a nice parameterization. Thenub314 (talk) 07:58, 3 February 2009 (UTC)

[edit] Possibly an error in sphere volume equation?

The last line looks like invalid, shouldn't the right side be something like: 2/3*pi*R^3 and not 2/3*pi*R^2 ?

Also not understanding where the divisor '3' appears to the right side.