Objective:
A physical demonstration that involves the approximation of an integral
with handson 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 nonstandard 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.
STANDARD 12oz MUG

Height = 5 in

Radius = 1.17 in

VOLUME = p
R^{2} H = 21.65 in^{3}
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.