Remember when I originally posted this problem back in January? Look here.

Here is the original problem:

A hole is drilled (bored) completely through a solid sphere, symmetrically through its center. If the resulting "hole" is 6 inches in height (or depth), show that the remaining volume must be 36π inches cubed.

OBJECTIVE: Motivation, explanation and application of method of cylindrical shells for finding volume of solid of revolution

TOTAL LENGTH: about 45 min

Please Note: These videos are not intended for students who want quick simple explanations for standard homework or typical exam items. This problem is above that level and the explanations are lengthy and very detailed!

Well, this 'video' is fragmented into 7 parts, the transitions are amateurish, it was composed over a few days (therefore different outfits!), cheap props and the quality is well, you know...

In spite of all the negatives, I'm hoping someone will find this helpful. Remember I'm doing this to cover a broad audience -- the Calc I/II student who wants understanding and clarity (not skipping steps!) to the AP student/Math-Sci-Engineering major who wants some theory and rigor. I'm also demonstrating some aspects of pedagogy here for the new calculus instructor who may have to prepare a similar lesson.

As mentioned in the video, there are many wonderful websites and videos which will provide better graphics, animation and quality. A couple of links are provided below. However, my purpose here to provide a highly detailed development of a classic calculus problem which reviews the method of cylindrical shells for volumes of solids of revolution.

Finally, my original intent was to find the volume that was removed by at least two methods and to generalize to a hole of depth h, but this is way too long as it is! Of course, I don't expect many views or comments but it will be out there for anyone who might have use of this for as long as this blog exists! I'm really hoping comments will look past the low-tech aspect and address the content and pedagogy.

Instructors

Please feel free to share this with your students or for whatever purpose you may have.

As stated above, the total length of all parts is about 45 minutes, the length of a typical hs class period, so it wouldn't make sense for the classroom. You might want to recommend students view this after learning the basic idea of the 'shell' method as reinforcement or after assigning this problem for hw or extra credit. These days students are savvy enough to locate, on the web, solutions and videos to most any problem we assign, so be careful! (You already knew that!)

Some Recommended Links

Volumes of Revolution - Cylindrical Shells

As mentioned in the video, patrickJMT is as good as it gets for clear, simple and mathematically accurate explanations.

Volumes - Cylindrical Shell Method

Wonderful explanations and excellent graphics and animation of the shell method (in Flash) from one of the best calculus sites on the web - utk (U Tenn Knoxville)

There are many other outstanding sites - I apologize in advance for omissions here. Just keep searching until you find the one that works for you!

As always, I am responsible for any errors - don't hesitate to point them out! At least we made it before XMAS 2008 ended!

The videos below are connected, so you might want to watch them in sequence.

However, the actual solution to the problem starts in the 5th segment below.

Read the descriptions of the segments to guide you in deciding where to begin. If you do not want a lengthy introduction, and already know the shell method, skip down to the 5th clip.

These first two video clips provide an overview for what I intend to cover.

Also the key relationship R^{2} - r^{2} = 9 is developed.

These next two segments motivate and derive the method of cylindrical shells.

The actual solution to the problem starts below!

Yes folks I know how drawn out this all was. I will try to improve on these but I will take an hiatus from my busy movie production schedule for awhile!

Happy New Year!

## Thursday, December 25, 2008

### Boring Hole in Sphere Calc Video - Finally!!

Posted by Dave Marain at 2:18 PM

Labels: calculus, hole in sphere problem, method of shells, video lesson, volume

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## 3 comments:

You can't get them all in one clip?

And what's with the football joke?

Anyhow, R asks me a related question - how quickly the contents of a spherical tank is rising if we know the rate at which the contents are being introduced.

I like the contrasting methods. But more than that, I like the voice. Really. Reminds me of who I learned from. And that's the voice I play in my head (it was actually as I was stuck in T-day traffic in CT) And strange the the answer was more dependent on the height of the contents than on the radius of the sphere... which made me think...

Anyhow, doubling back. If you are doing these as a resource, you need to find a way to put them in one clip.

I don't know if I could learn first time from this... but if I'd seen it once, somewhere else, this would be perfect.

Afa the technology, there might be ways to cut back and forth to computer graphics... but I'm not volunteering. I can't even really do diagrams!

Thanks!

Jonathan

Thanks, Jonathan!

You're right -- way too fragmented to be an easy learning tool. I could get most of it into 1-2 video clips but the quality degrades quickly from the double compression it goes through. If I were doing short simple exercises this would not be an issue but there are other sites doing those. I really wanted to do an in-depth calculus problem which demonstrates several instructional strategies and combines concept and application.

Thanks for the 'voice' comments! I'm a frustrated singer although my wife says I harmonize with the vacuum cleaner! Seriously, I guess I miss the classroom so a lot of this is my connection to that.

As far as the Jets shirt is concerned, since I've been wearing it they haven't won very much! I really do like both the Giants and Jets just as I like both the Yankees and Mets. It's either schizophrenia or just the Libra in me.

Thanks again for the support. I'll think about that related rate/sphere problem...

Thanks to Jonathan's comments and my own review of the post, I've made some changes:

(1) Restatement of the original problem

(2) Preview of the video clips so that you can decide which ones to view

Enjoy!

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