Consider a 5-hour train ride that takes you 500 miles directly away from your home, in a straight line, to another destination where you stay for 5 hours and then return. Suppose you want to stay wirelessly glogged into your home computer at a "roaming" communications cost of $1/mile/hour. For simplicity, assume a linear long-distance rate, i.e. $1/hour when you're 1 mile away, $2/hour when you're 2 miles away, $3.14/hour when you're 3.14 miles away, etc.. The total cost of your online communications is $5000, since the absement (time-integral of displacement) is 5000 mile hours (1250 mile hours on the way to your destination, plus 500 miles * 5 hours stay = 2500 mile hours, plus 1250 mile hours of absement during the return trip).
The middle plot shows Displacement. The first 5 hours are spent in the
train going at velocity 100mph (miles per hour) away from home.
The area under this triangular part is 1/2 five times 500 mile hours,
which is 1/2 times 2500 mile hours, i.e. 1250 mile hours.
The next 5 hours are spent at your destination, 500miles from your home,
where you pay $500/hour for 5 hours, for a cost of $2500.
Staying online during your return trip costs you another $1250.
Your total cumulative running cost is the area under the middle plot up to a particular point in time. This integral is called absement and is shown on the top plot.
Each of the three plots is the time-derivative of the plot above it:
Hydraulophone in Experimental Musical Instruments publication
See and hear a musical composition for hydraulophone and orchestra:
Here's a paper on teaching absement to children,
published March 13, 2008, by Harbourfront Centre in Toronto:
Harbourfront_centre_Hydraulophone_Absement_Theory.pdf