Confusion about PSD and RMS
I think what your missing is that the Power Spectral Density, as a density is the power per unit frequency, so assuming units of Hz as is typically done, $\sigma^2/f_1$ in your case is the power per Hz; and therefore to get the total power, $\sigma^2$, you would multiply that by the total frequency $f_1$ resulting is the expected $\sigma^2$.