Yes.

Yes.and isnt

c'(x)(the derivative of c) the function of the curve above the two

regions(i.e region under f1(x) and region under g1(x))

between a and b?

Yes, to all of those questions.Or at least if the function of the curve above the two

regions(i.e region under f1(x) and region under g1(x))

is fn(x)

isnt fn(b)=c'(b);

by second fundamental theorem of calculus

would be greatful for any reply.thanks