Idea: Integration by parts.
Prove that if f&g are continous and inverse functions for each other and a&b are constant where b>a .. then :
My FAILED try:
Am thinking about a substitution which makes the f(x) be g(x)
So I substitute x=g(g(x)) ..
But this failed; since dx will be compliacted
I think it works for f increasing...
... but not decreasing...
I.e. subtracting as directed in your formula leaves the white region(s) inside the larger rectangle - which correspond(s) to
in the increasing case only.
Edit: on the other hand...
Ah! Should have tried integration by parts before sounding off...
Still bothered by my graphs, though...
Late edit: Thanks for exuming this, BayernMunich, but the less said the better! I did spot my VERY SILLY graph error in the end!