Greetings MMD,  Since tracker tubing comes in 100 foot rolls and since
an upright is going to use more than one roll, a precise formula would
probably not be of much practical use.  The curve of the tubing as well
as other factors would enter into an exact formula.  For practical
purposes it make sense to me to just use the following:

  total length = (shortest + longest)*(number of tubes / 2)

In other words, the average of the longest tube and the shortest times
the number of tubes.  This would only be strictly accurate for straight
lines of tubing.  Variations of the curve of the tubing would bias the
average if the distribution of the variation were not uniform but I
doubt it would be off by more than a few feet for the average piano.

Another approach would be to take all the old tubing off and weigh it.
Then weigh a 10 foot length of the old stuff and divide the total
weight by this.  The result would be the number of ten foot lengths
needed.  Divide that by 10 and get the number of 100 foot rolls needed.

Best regards,
Spencer Chase
Laytonville, Calif.
http://www.spencerserolls.com/