Volumes of Solids of Revolution

 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.