theorem :: INTEGR12:40

for a being Real
for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & g = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * (((a ^2) - (x #Z 2)) #R (- (1 / 2))) & f1 . x = a ^2 & g . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * g) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * g)) . (upper_bound A)) - ((- ((#R (1 / 2)) * g)) . (lower_bound A))