The convolution-type Volterra integral equation of the first kind
$f(t) = \int_a^t k(t-t')\,x(t')\,dt' \qquad t\in [0,\infty]$
can be solved (at least formally) by applying the Laplace transformation $L$ to obtain
$F(s) = K(s)X(s)$
(where the upper case denotes the Laplace transformed lower case function) and then applying the inverse Laplace Transform $L^{-1}$ to obtain
$x(t) = L^{-1} [F(s)/K(s)]$.
My question is: Can this method of solution be modified to applying to $f(t)$ which are defined only on some finite interval $[0,T]$?