The Fredholm equation of the first kind is $$ f(x)=\int_a^b K(x,t)\phi(t)\, dt \tag 1 $$ Q1:
Does it mean $x$ is a scalar $x\in \mathbb R$?
Or is it a function $x:\mathbb R\rightarrow \mathbb R$, i.e. $x(t)$, so explicit we have $$ f(x(t))=\int_a^b K(x(t),t)\phi(t)\, dt \quad ? \tag 2 $$
Q2:
And the same question for the Volterra equation of the first kind. Is it $$ f(x)=\int_a^x K(x,t)\phi(t)\, dt \tag 3 $$ Or $$ f(x(t))=\int_a^{x(t)} K(x(t),t)\phi(t)\, dt \quad ?\tag 4 $$
Q3:
What about the Fredholm equation of the second kind and the Volterra equation of the second kind?