Q:

How can I call the area under an impulse/spike?

I am looking for an appropriate term to express the area under a spike or impulse for time series signals. The term "Briefness" is probably not appropriate since the amplitude should be taken into account

A:

What you're describing "literally" is just the area under a curve, the integral. I don't think this helps you.
What you want, functionally, is not quite clear, but I'd bet it's either

the energy of the pulse or
the energy-bandwidth product.

The energy of the signal is the integral over the (magnitude)
squared signal in the continuous time case, or the sum over the
(magnitude) squared signal in continuous time.
The bandwidth is a frequency domain concept, and will tend to be larger the "shorter and sharper" the pulse is. The product of bandwidth and energy has important estimation quality effects for example in radar applications.

