Approximating Volumes of Solids of Revolution
  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits


    Objective: A physical demonstration that involves the approximation of an integral with hands-on measurements.

    Level:  Second Term Calculus or the appropriate course where volumes of solids of revolution and approximation of integral are discussed.

    Prerequisites: Solids of revolution have been discussed and several were computed using appropriate integrals. Simpson's Rule has been illustrated for functions when estimating area under a curve. 

    Platform:  None.

    Instructor's Notes:  Students often have trouble estimating the volume of a container that has a non-standard shape. Much of the time their estimates for the volume of 'My Favorite Mug' shown below are much too low. The developments of volumes of solids of revolution and Simpson's Rule for approximatin integrals are combined to obtain a quite accurate measure of the volume of such mugs.

    After the mugs are displayed to the class, information about the 12oz mug and leading questions about My Favorite Mug can be incorporated into a discussion.
            STANDARD 12oz MUG
            • Height = 5 in
            • Radius = 1.17 in
            • VOLUME = p R2 H = 21.65 in3
            MY FAVORITE MUG
            • Guess my Volume.
            • Estimate my Volume using Physical Measurements.

      Performing the Estimation:

      • Using a measuring tape determine the circumference of 'My Favorite Mug' at the top, middle, and bottom.
      • Compute the radii of the corresponding cross sections.
      • Use Simpson's Rule to estimate the volume.
        If y = f(x) is the equation of the curve that generates 'My Favorite Mug,' then

      > A Check: Fill MY FAVORITE MUG with water and measure the volume.
      > Compare the measure with the approximation; discuss possible reasons for discrepancies.

      The estimation procedure is shown as an animation as follows.

      Credits:  This demo was submitted by 

      Dr. Klaus Volpert
      Department of Mathematical Sciences 
      Villanova University

      and is included in Demos with Positive Impact with his permission.

    DRH 10/25/99   Last updated 5/23/2006

    Since 3/1/2002