Let be $X(t)= x - μt + σB_t - N_t$ a L$\acute{\text{e}}$vy process starting from $x >0,$ where $ μ\ge 0, \ σ\ge 0, \ B_t$ is a standard BM, and $N_t$ is a homogeneous Poisson process with intensity $ θ>0,$ starting from zero. We study the joint distribution of the first-passage time below zero, $τ(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $τ(x).$ In particular, we establish differential-difference equations with outer conditions for the Laplace transforms of $τ(x)$ and $A(x),$ and for their joint moments. In a special case $(μ= σ=0),$ we show an algorithm to find recursively the moments $E[τ(x)^m A(x)^n],$ for any integers $m$ and $n;$ moreover, we obtain the expected value of the time average of $X$ till the time $τ(x).$