  We consider the process $\{x-N(t):t\geq 0\}$, where $x\in\mathbb{R}_+$ and
$\{N(t):t\geq 0\}$ is a renewal process with light-tailed distributed holding
times. We are interested in the joint distribution of $(\tau(x),A(x))$ where
$\tau(x)$ is the first-passage time of $\{x-N(t):t\geq 0\}$ to reach zero or a
negative value, and $A(x):=\int_0^{\tau(x)}(x-N(t))dt$ is the corresponding
first-passage (positive) area swept out by the process $\{x-N(t):t\geq 0\}$. We
remark that we can define the sequence $\{(\tau(n),A(n)):n\geq 1\}$ by
referring to the concept of integrated random walk. Our aim is to prove
asymptotic results as $x\to\infty$ in the fashion of large (and moderate)
deviations.
