REPLY:
A TYPICAL EXPLANATION I'D GIVE:
1. (rhetorical question) Why is there any contradiction between infinite
(2D) area and finite (3D) volume?
2. (followup) After all, the 1D "length" of the interval (0,00)
("00"=infinity) is infinite while the 2D area under the exponential function
on (0,00) is finite. (...and 'the length fits inside the area'?)
SEQUEL:
The way I've heard the original dilemma posed is that finite paint will fill
the "infinite can" BUT finite paint cannot cover its inside surface.
A way out of this apparent dilemma comes from realizing that the paint
canNOT be of (any) uniform thickness because the limit of the "can's"
diameter=0.
I think these approaches are valid and also accessible to Calc 2 students.
-------------------------
Professor Ken Constantine
Eastern Nazarene College
Quincy, MA 02170
(617)745-3501
Email: constant@enc.edu
------------------------
edmund sears, 3'rd verse of "It Came Upon A Midnight Clear":
"And ye, beneath life's crushing load, whose forms are bending low
Who toil along the climbing way, with painful step and slow,
Look up! for glad and golden hours come swiftly on the wing.
O rest beside the weary road and hear the angels sing."