for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / 2) (#) ((#Z 2) * arccos)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccos)) . (lower_bound A))