Sample and hold - LTI filter
An impulse of "area" $A$ means $A\delta(t)$. The nomenclature arises because
$\displaystyle \int_{-\infty}^\infty A\delta(t) \,\mathrm dt = A,$
that is, the "area under the curve" is $A$.

Since the system is linear and
time-invariant, and the response to input $\delta(t)$ is a pulse of height $1$
and duration from $0$ to $T_s$, the response to $A\delta(t)$ is a pulse of
height $A$ and duration from $0$ to $T_s$.  Its "area" is $AT_s$, that is, $A$
multiplied by the area $T_s$ of the response to the unit impulse.