Does the value of a, when a does not equal b or a is greater than b, influence the result when using FTC 2?

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the fundamental theorem of calculus, part 2 is derived using the assumption we are using the closed interval [a,b], so that a≤x≤b However, see http://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx#Extras_IntPf_Def1 the 3rd and 4th proofs show the effect of a>b or a=b (i.e. in the first case we get the negative value of integrating in the "normal order" small to large, and in the second case a=b, we get 0 for the integral