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					              Slabs to Volumes: Classic Calculus




The Egyptians and the Babylonians knew the formula for pyramid and truncated
pyramids
                                   1
                                     (base)(height)
                                   3

 The formula is proved in Euclid’s Elements. We have a few other formulas for
volumes, but not many. Part of the power of calculus (and algebra) is that we may
approximate the volumes of many weird shapes using successively more accurate
applications of the idea we’ll explore this morning: slicing.

Question
How do calculus ideas lead to a value for the volume of a right pyramid with a square
base (4 cm by 4 cm) and height 2 cm? To start, in this special case we have the Egyptian
formula to check our calculus. It says that the volume is _________ .




Part I: Thought Experiment

   1. Imagine slicing the pyramid as you have sliced the egg. Because your imagined
       pyramid is a “cleaner” mathematical object, the slices you see in your mind’s eye
      should be more regular than our “rounded” egg slices. Sketch one of the slices,
      and name the shape it has from the side and from the top:
   2. Imagine approximating each of these slices with a slice whose faces are all
      rectangular. Sketch one slice, and then sketch your imagined rectangle
      superimposed on the slice. How can you make the rectangle approximate well?




   3.    Read and reflect on the following notions from calculus use slices like ours to
        model volumes. The font we use is called party: these ideas are so fundamental
        to this part of calculus that we should celebrate as we digest and see how they
        connect to the egg experiment and to our being about to avoid dipping everything
        whose volume we want to find.

         Knowing the thickness of each
          rectangular slice we can computer each
          slice’s volume because this 3D shape
          has a “known” volume formula

         The sum of these slice volumes
          approximates the pyramid volume.

         If the slices aren’t too thick, when
          they are stacked up they even look like
          a pyramid (jagged profile overlooked).

Are you reminded of the Riemann rectangles from the area work?


Part II: Construction Experiment

We shall construct a stack of similar shapes. If the layers are thin enough, it might look
like a pyramid. For thicker layers, the side view will show jagged edges.

Materials:
   cardstock, cardboard, or better yet, sheets of foam, to make your layers.
   scissors or something to cut your material
   some sort of adhesive to paste them all together
   a rule to measure edges and heights.

Design & Record
   1. Measure the thickness of your stacking material. For instance, my cardboard
      model used cardboard 1/4 cm thick.

   2. Design successively smaller squares that they “stack” into a rough pyramid. Be
      systematic in your design: shorten each side by some fraction that divides 2
      evenly, e.g. 1/4, 1/8, 2/5. We’ll want to algebra-tize your process.

   3. Complete the table on the next page with the dimensions of your slabs. The
      number of slabs you need depends on the thickness of your cardboard

                              Slab # Height Side length of Base Slab Volume
                                 1    1/4          4 cm
                                 2
                                 3
                                 4
                                 5
                                 6
                                 7
                                 8
                                 9
                                10
                                11

In hopes of finding a computational pattern, complete the following statements about the
approximating sum of slab volumes. Compare this to how we symbolized the
approximations to egg slice volumes. For a pyramid with 13 slabs or layers, each of the
13 term is a volume of a “slab” a.k.a. square cylinder, square prism or parallelepiped.

      Each term is a product representing (base area)*height as with any cylinder
      ¼ is present because all cardboard slabs have this height
      parentheses enclose factors giving increasingly smaller and smaller lengths of the
       base edges

Total Volume = volume bottom slab + volume slab 2+ ….+ volume top slab
                       top
              =         volume slab 
                  k bottom
                                           k




                                                ...      
                                       2           2               2
              =   1
                  4
                       42  1 
                             4
                                               1
                                               4
                                                           1
                                                           4



              =   1
                  4
                                                                     
                                                                 2

                                                                
                                13
                    =   1
                        4
                            
                            k 1


You’ve seen a variety of number patterns: have you toyed with the pattern of summing
squares? Let’s sum squared halves just because that’s what we have to work with in the
volume sums:
    2           2           2                    2
 1   2   3  4  1                    1 656 1 459
            1  4  9  16  
 2  2  2  2
                                                   
                       4                    4 6      4 6
    2           2           2                    2           2
 1   2   3  4   5  1                         1 5  6 11
               1  4  9  16  25  
 2  2  2  2  2     4                         4     6
    2       2                       2                2
 1  2          6  7  1                      1 6 140 1 7  8 15
      ...        1  4  ...  49  
 2  2          2  2
                                                            
                             4                      4 6       4    6

Do you see a pattern emerging? Try the sum representing my cardboard pyramid:

                                                         2
                                         k13
                                1
                                         1                            1
                                4
                                         2
                                        k 1
                                              4
                                                                     4*6     4




Thinner Slabs & More Accuracy

Either imagine or construct slices that are half as thick.

Q1: How many more slices to make a pyramid the same height will be needed if layers
are half as thick?


Q2: How will the above sum for the volume change with halved slices?




Algebra-tizing
The key to using calculus is the same as the key to using your calculator to help you do
calculus: generalize a number pattern. Just as our work on our area and length paradoxes
involved “infinitely repeated” smaller blocks or steps, this computation involves adding
more and more, smaller and smaller squares. It’s this many, many smaller and smaller
pieces that is synonymous with integral calculus.
Q3: How would you change the summing expression for your slabs? Discuss the
process of better approximate volumes when the slices are thinner.

				
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posted:9/15/2011
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