I want to report statistics for a region of interest $[x_0, x_1)$ of a function $f(x)$. As a concrete example, consider $f$ to be the spectrum seen by a radiation detector, such that $x$ has units of energy in keV and $f$ has units of "counts/second/keV". (Ignore for the moment that such spectrum would normally be discreetly-binned values rather than a continuous function.) Basic quantities I can produce are:
- Integral [counts/second] = $$\int_{x_0}^{x_1} f(x)\ dx$$
- Average [counts/second/keV] = $$\frac{\int_{x_0}^{x_1} f(x)\ dx}{\int_{x_0}^{x_1}dx} = \frac{\int_{x_0}^{x_1} f(x)\ dx}{(x_1-x_0)}$$
- Mean [keV] = $$\frac{\int_{x_0}^{x_1} x\cdot f(x)\ dx}{\int_{x_0}^{x_1} f(x)\ dx}$$
- ??? [keV/second] = $$\int_{x_0}^{x_1} x\cdot f(x)\ dx$$
Generically, is there a proper term for quantity (4), which is constructed as an unnormalized mean or equivalently the mean times the integral? Continuing my concrete example, this quantity has useful physical meaning as either the radiation "dose" or the "power" absorbed.