Lesson #110 Volumes of Solids of Revolution (Disk Method)


Quote of the Day:
"One can invent mathematics without knowing much of its 
   history.  One can use mathematics without knowing much, 
   if any, of its history.  But one cannot have a mature 
   appreciation of mathematics without a substantial 
   knowledge of its history."  -- Abe Shenitzer 

Objectives:
The student will compute the volume of solids of revolution
   using the disk method (slicing).

Materials needed:
Styrofoam disks of various sizes along with a dowel.
Power Drill and safety glasses.
Styrofoam cut-outs mounted on dowels.

1. Collect homework.  

2. Recall that to find the area of a plane region, we 
   divide the region into thin rectangles, add the areas of 
   rectangles to form a Riemann sum, and then take the 
   limit of the Riemann sums to obtain an integral for the 
   area:

       

3. This week, we are going to examine volumes of solids of 
   revolution.  We use the same strategy to find the volume 
   of a solid.  We will divide the solid into thin slabs, 
   approximate the volume of each slab, add the 
   approximations together to form a Riemann sum, and then 
   take the limit of the Riemann sums to form an integral 
   for the volume of the solid.

       

   In our first method, we will be summing up volumes of disks (or cylinders).  

       

       

   Show Styrofoam circles on a dowel to illustrate this idea.

4. Example

       

       

       

5. One of the most difficult things to do when working with 
   volumes of solids of revolution is to visualize the 
   shape that is being formed.

   To help with this visualization process, use one of the 
   following techniques:

   (A) Use styrofoam disks
        Click here for a picture

   (B) Use power drill with Styrofoam cut-outs mounted on 
       dowels.  When the styrofoam rotates it traces out 
       the solid of revolution.
        Click here for a picture

   (C) Do some edible calculus (by Nancy Dirnberger): (USE after SHELL METHOD)
       If you core an apple you have a great solid of 
       revolution with a hole through the solid! An apple  
       is usually sliced in one of two ways. If you slice 
       it so that the plane of the slice contains what
       would be the axis of revolution, your apple slice 
       (actually you have two) is a disc of revolution. You
       can almost slice it thin enough to have width dx! If
       you slice the apple into rings, the resulting slices
       are washers that will give you the same volume when 
       added.  With a large, thick slice of a Bermuda onion 
       you can pull up onion rings as successive 
       cylindrical shells. Students often have a problem
       visualizing these cylindrical shells.

6. Determine the volume of a sphere.
       

7. Example (with respect to the y-axis)

       

8. Assignment:
   p. 456 (1, 3, 7, 10, 13)  

   Worksheet "Dorothy and Kansas" Problem (PDF File)         

   Worksheet "Dorothy and Kansas" Problem (Word DOCUMENT)
Lesson #110 Volumes of Solids of Revolution (Disk Method)

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