Apply band pass filter (BPF) before calculating the power of a signal (PSD)?

From my understanding PSD tries to give a good estimation on the power each frequency attributes to the overall signal power.

If I am only interested in a frequency range $(\omega_0 \leq \omega \leq \omega_1)$, could I put the signal through a band pass filter for that same frequency range then calculate the signal power using this formula:

$$P_x = \frac{1}{N_1 - N_0 + 1}\sum_{n = N_0}^{n=N_1} \left|x(n)\right|^2 $$

Would this technique give me what I am looking for, (the power of the signal from $N_0$ to $N_1$ for the specific frequency range)?

How does this compare to PSD estimation using any of the popular techniques?

Pros, Cons?

Thank you!