Calculating power in frequency band using power spectral density
It's less about power spectrum and more about Fourier transform of real signals. Let's take an easy signal, $g(x) = \cos(10\pi x)$.

Then:
$$ F(\omega) = \frac{\delta(\omega -10) + \delta (\omega +10)}2.$$

When you ask "what is the energy in the band $ \omega = [9,11]$ you ask what is the energy of that cosine (of course this is an example).

If you look only at the positive side of the frequency you lose half the energy.