# Talk:Volume of Revolution

## Contents

#### Abram 7/9

Nice job on the edits. Most of the issues you addressed completely. I would say that 2 of the issues are mostly resolved, but not entirely.

The first unresolved issue is that I realized your whole explanation is specifically addressed at revolving functions around the x-axis. If you revolve around the y-axis, for instance, you would have to use delta-y slices if you wanted to use the disk method.

There are two ways to address this issue. The easier, and not bad way, to deal with this is to replace "fixed axis" with "x-axis" in the sentence "given a function, we can graph it then revolve the area under the curve between two specific x-coordinates about a fixed axis to obtain a solid called the solid of revolution."
The less easy, but slightly better way to deal with this is to make three changes. First, in front of the sentence quoted above, put another sentence that reads something like, "In general, a volume of revolution can be formed by revolving any plane area, bounded by curves of almost any form, around any fixed axis. The calculations can get difficult, though, so we will focus on a specific case." Second, do the replacement described in the previous paragraph. Third, replace "delta x" in the first paragraph of the Basic Description with "very thin slices."

The second issue is that your extension of the bread analogy is 90% of the way there, but it would still help to explain what the value of y corresponds to in the slice of bread. This, in turn, will probably require you to explain that the differences between the loaf of bread and the solid of revolution are that the bread slices have roughly square cross-sections, while the solid has circular cross sections. I really like your idea to keep referring to the bread -- I just think you should push it a bit farther.

#### Abram 7/7

Lots of good content on this page, as always. A few more suggestions.

First, two things regarding images:

• I would actually take out the animation you left in and put back in the one you took out. I don't think the animation that is in there actually helps that much. I'm not quite sure why. Maybe because nothing (like axes) are labeled, so it's hard to tell there's any coordinate geometry going on. Anyway, I'm pretty confident that the image you took out is actually the more useful one.
• I still think an image showing the y=x^2 paraboloid broken into slices, with y and delta x labeled on one of those slices, would be incredibly helpful, though I also know that's a lot of work.

Also, a couple issues regarding text:

• The basic description should have "about the x-axis" added to it.
• Where you write "Volume of disks = [summation formula.] This is the same as [integral]", you need to insert something like, "If we make the disks infinitesimally thin, this is the same as [integral]." Without some kind of transition like this, what you have written isn't actually true.
• There are a few problems with the text that reads:
If we are given a function which decribes the shape of a solid, we can plot the function, then revolve the resultant plane area about a fixed axis to obtain the original solid, now called the the solid of revolution. The area can be bounded by many curves of almost any form.
• What resultant plane area? I'm guessing you mean the area under the graph of the function that is between two given x-values, but could you clarify?
• If I'm right about what you mean, this description doesn't jive with the sentence "The area can be bounded by many curves of almost any form", which refers to a much more general case. The rest of your page really just describes finding the volume when the plane area is as described above, so maybe you could just delete the sentence about "the area can be bounded..."

Sorry, I know this is a lot of suggestions, but except for the one about the image I think should be added, these should be pretty quick.

#### Lizah 6/28

Hey Anna,

//images that are not related to your example by the text of your example. Is there a way to fix that?//

I tried fixing this by giving a small explanation for the images to show why they are relevant and related to the example. Though they do not necessarily tie with the example, they help illustrate the general idea i'm trying to explain. About the formatting issue, i'm still trying to figure that out. I'll definitely talk to Maria about it to see how we can position the images so that they don't look like they are littering the page.

Thanks for you suggestions Anna

#### Anna 6/26

Hi Lizah,

I've thought about your formatting problems (too many images in too small a space). Could you talk to Maria or Keith about making sure that all of your smaller pictures are actually below the big one? I think that would help a lot, and make it easier to break that first paragraph up into smaller pieces.

I also am finding that there are images that are not related to your example by the text of your example. Is there a way to fix that?

#### Abram 6/15

Hi Lizah,

Thanks for your replies to our suggestions and for implementing some of the changes I suggested. First, I'll reply to your replies:

//I'm not sure what you mean by this Abram 'cause the title of the page is "Volume of Revolution" and my first sentence starts with "When finding the volume of revolution...."//

You're totally right. My bad. However, I just realized that what would be great would be an introductory sentence about what a volume of revolution is. An easy way to accomplish this might be to open your basic description with the definition of a solid of revolution that you have a couple sentences down.

///Also, is it possible to show an image of the disk method that has things like y, delta x, etc, labeled, and maybe even a derivation of the volume of one of those disks?///
//I considered doing that, but I thought that the page would be littered with too many images. I'll talk to Maria about it to see if I could change the page layout so that I can fit one or two more helpful images without making it look too much. //

I understand that concern. I would say that you can definitely drop one of the two animated images. Animations are lovely, but this kind of image is so helpful.

///I really like your first main paragraph, but I do think that it would be easier to read if you broke it up into several shorter ones. ///
//I'm having a little trouble with that because of the number of images I have. I talked to Maria about it to see if there is a way we could format the page so that I can have shorter paragraphs and all the images in without leaving too much space. //

Gotcha.

OK, on to other things.

//If you are given a function which decribes the shape of a solid, plot the function, then revolve the resultant plane area about a straight line to obtain the original solid, now called the the solid of revolution.//

This is a great description which can use some cleaning up.

• Generally, a solid of revolution is obtained by taking any area in the plane and revolving it around a fixed axis. This area can be bounded by many curves of almost any form. What you are talking about for almost the entire page is solids obtained by revolving the graphs of functions of x about the x axis, so it may help to restrict all explanations to this case, and just note somewhere that you are doing so.
• What do you mean "which describes the shape of a solid"? You could strike this completely, and just say "given a function"
• "plot the function" should be "plot a graph of the function" or "graph the function"
• Be clear about what plane area you are talking about (i.e. the area between the graph of the function and the x-axis or the "area under the curve" with a link to a helper page on area under the curve). Note that this definition in particular implements the restriction cited above.

- I would suggest boldfacing the first use of "disc method", because you are about to define it.

- The Riemann sum bit will have to be reworded somewhat, as a Riemann Sum does not by definition imply infinitesimally thick slices. I'll think about how to reword this, or you can think about it, or you can talk to Steve or Gene.

#### Lizah, 6/15

Hey Alan, Abram and Anna

Thank you all for the helpful suggestions you guys shared. I effected most of the changes you guys recommended.

//You could dance around all of those issues by explicitly making it a helper page//

It is an image page. Alan, you might be confusing it with the differentiability page, which is a helper page.

//The page is not called "finding the volume of a solid of revolution", so it may help to make the first sentence of your basic description say that you are going to write about how to find the volumes of these solids (as opposed to looking at anything else about these solids). //

I'm not sure what you mean by this Abram 'cause the title of the page is "Volume of Revolution" and my first sentence starts with "When finding the volume of revolution...."

//Also, is it possible to show an image of the disk method that has things like y, delta x, etc, labeled, and maybe even a derivation of the volume of one of those disks?//

I considered doing that, but I thought that the page would be littered with too many images. I'll talk to Maria about it to see if I could change the page layout so that I can fit one or two more helpful images without making it look too much.

//I really like your first main paragraph, but I do think that it would be easier to read if you broke it up into several shorter ones. //

I'm having a little trouble with that because of the number of images I have. I talked to Maria about it to see if there is a way we could format the page so that I can have shorter paragraphs and all the images in without leaving too much space.

Thanks guys

#### Alan, 6/12

About the discussion on the opening sentence and the relevance to the main image:

You could dance around all of those issues by explicitly making it a helper page. Right now it looks like it is in the image page format. If it was a helper page explicitly there wouldn't be a need for a main image or that introductory sentence about the image.

#### Abram, 6/12

Hey Lizah,

I really like your basic description of how the disc method words, and the example you chose is great because it's just complicated enough to warrant using a disk method without being so complicated as to distract from the point. Many of the images you included are really helpful.

In the introduction your description of "finding a function which describes..." is a little bit confusing because finding a volume of revolution usually begins with being *given* one or more functions, and asking what the resultant volume is when the area enclosed by the graph of those functions is revolved about an axis.

The page is not called "finding the volume of a solid of revolution", so it may help to make the first sentence of your basic description say that you are going to write about how to find the volumes of these solids (as opposed to looking at anything else about these solids). Similarly, you may want to highlight that the disk method is one way of finding the volume of such a volume, but that there are other ways to compute the volume, rather than simply dive into using the disk method.

All these suggestions may help the reader be a little clearer on exactly what kind of problem you are going to solve and where you are headed at any given moment.

Also, is it possible to show an image of the disk method that has things like y, delta x, etc, labeled, and maybe even a derivation of the volume of one of those disks?

Hey Lizah, I've got a couple of things to point out. The first is that your image of the "area" really just as a line. If you created it in mathematica, you can go back and use the "filled" command. (searching help will get you information on the specific syntax.

I really like your first main paragraph, but I do think that it would be easier to read if you broke it up into several shorter ones.

I might also make the computation something that you can hide, instead of a completely separate page.

Another thing to do is possibly show (and have a hide option) for comparing the volume you find with volumes found using other approximation methods (ie, treating the whole shape like a cone).

-Anna 6/11