Upper and lower limit Calculus problem.

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In the book which I am reading/understanding, there is an integral(actually it is part of gauss' divergence theorem proof),

$ \int_{f_1(x,y)}^{f_2(x,y)}\frac{\partial F_3}{\partial z}\;dz $

Now, they have solved it this way,

$ =F_3\{x,y,f_2(x,y)\}-F_3\{x,y,f_1(x,y)\}\;\;-(i) $

Now, I am not able to understand how they did it. Till, now, I have learnt to directly put the upper end and lower limit. For example,

$ \quad \vert x\vert_1^2 = 2-1;\;\;-(ii) $

But, in the above case eq(i), they have somewhat not proceeded like eq(ii), Now can anyone please explain what exactly is happening?