Boring Hole in Sphere Calc Video - Finally!!

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 R² - r² = 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!

A Holiday Geometry Gift -- All Rolled Up For You!

Whether you view this as an SAT-type problem, a geometry challenge or just another investigation, I hope you will enjoy this in the true holiday spirit of giving! So Happy Chanukkah and Merry XMAS!

OVERVIEW

Math Standards

• 3-dimensional objects, spatials sense, volumes of cylinders
  

Target Grades:
Although this investigation appears to be aimed at secondary students taking geometry, cylinders are introduced in middle school and even earlier. There's no reason why middle schoolers shouldn't be able to tackle this or perhaps a modified version. Are 7th and 8th graders expected to know the formula for the volume of a cylinder, or at least, the general form: Area of Base x Height?


THE PROBLEM
In each figure, a 3x5 rectangle is rolled to form a cylindrical shape (a "can" without a top or bottom). In Fig. I, we "identify", i.e., paste edge AD onto edge BC. For Fig. II we reorient the same rectangle and paste edge DC onto edge AB. Even though these cylinders do not have a 'bottom', assume they are sitting on a flat surface and we will be "filling them up" to determine their volumes.

For the Instructor: Provide 3x5 index cards for the students. Students, working in pairs, should physically form each of the cylindrical shapes and refer to these while working the problem.

Investigation/Questions

1) Without calculating, make a conjecture: Which cylinder would have the greater volume or would they be equal? (For instructor: Record the results of these conjectures on the board)

2) Compute the volumes of each cylinder - no calculators! Leave results in terms of π. Was your conjecture accurate?

3) Compute the ratio of the volume of cylinder I to the volume of cylinder II. Any surprises?

4) It wouldn't be an investigation if we didn't generalize! But this time, YOU have to write the generalization, state the conclusion and prove it! Remember, the true spirit of the holidays is to give, not only receive!!

HAPPY HOLIDAYS
FROM
MATHNOTATIONS

A 'Boring' Volume Problem or "If You Find Yourself in a Hole, Stop Digging!"

Important Note: It took forever but I finally posted the detailed video explanation of this problem here.


Please don't gag on my feeble attempt at humor in the title (my wife actually had bought a sign with that quote -- it's hanging on the dining room wall).

There are a couple of classic volume problems in calculus which have always been my favorites:

• The Volume of the Torus Problem (using 2 methods: cylindrical shells and by disks)

• The Hole in the Sphere Problem (also by 2 methods)

I always assigned one or both of these to my BC Calculus classes, most often as Extra Credit problems. Because of the extra points they could earn, most students tried these and submitted solutions. My feeling is that if a student could do both of these by both methods, they really understood disk and shell!

In this post we will focus on the 2nd problem as it always seems to generate curiosity and interest. I'm guessing that most of you know the puzzle version of this question that was answered by Marilyn vos Savant in her Ask Marilyn column over a decade ago. It's just possible that some calculus student in some second semester class is feeling some anxiety over this problem!

Here's one version of that famous conundrum. There are many approaches here, even the clever mathematical approach of assuming that the problem is well-defined and therefore independent of the radii involved (I expect at least one of our readers to do it that way!).

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.

That's right, the answer is independent of the radius of the sphere and the diameter of the hole! The total volume of the sphere and the volume removed however do depend on the radii. Note that the volume removed is a cylinder with two spherical caps.


The original problem was worded ambiguously in Marilyn's column and then clarified somewhat. My version is not perfect but hopefully you'll get the 'picture', although a real picture would be far better. I will probably do a video presentation of the solution and a discussion of the problem because the diagram and the math expressions are cumbersome and it's not worth the time to play with Draw programs or LaTeX right now. I plan on presenting in detail the disk-washer and cylindrical shells method using a general depth of h inches for the hole.

For now, have fun playing with this. This is a well-known problem and therefore searchable on the web but try it yourself first. Try to use calculus to set up the integral and if you're brave you'll evaluate those integrals without Mathematica or the TI-89! Can you see why the answer for the volume remaining depends only on the depth of the hole?