  For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$
starting from $x<S,$ we study the probability distribution of the integral
$A_S(x)= \int_0 ^ {\tau_S(x)}X(t) \ dt$ determined by $X(t)$ till its
first-crossing time $\tau_S(x)$ over $S.$ In particular, we show that the
Laplace transform and the moments of $A_S(x)$ are solutions to certain partial
differential-difference equations with outer conditions. The distribution of
the minimum of $X(t)$ in $[0, \tau_S(x)]$ is also studied. Thus, we extend the
results of a previous paper by the author, concerning the area swept out by
$X(t)$ till its first-passage below zero. Some explicit examples are reported,
regarding diffusions with and without jumps.
