can the definite integral be considered as the output of the indefinite integral function, or is there a way to obtain this?

Question

How to define the link between indefinite integral and definite integral?

Indefinite integral is a function, while definite integral is a number.

What is this number relatively to this function?

Could one say that, for every definite integral of f(x)dx from a to b , there is an indefinite integral of f(t)dt from a to x, lets call it function A, such that :

(1) b belongs to the domain of A

(2) the definite integral is the image of b under A ?

I tried to give an example of what I intend to say with the image above.

Answer 1

If $f$ is an integrable function from $[a,b]$ to $\mathbb R$, then define$$\begin{array}{rccc}F\colon&[a,b]&\longrightarrow&\mathbb R\\&x&\mapsto&\int_a^xf(t)\,\mathrm dt.\end{array}$$Then $F$ is an indefinite integral (or a primitive) of the function $f$ and$$\int_a^bf(t)\,\mathrm dt=F(b).$$