As I am not a professional mathematician/statistician maybe someone of you can help me writing
the correct mathematical notation for following procedure/task:

Assuming the probability function like the standard normal distribution with  (\mu = 0, \sigma^{2}=1)
is f(x)

The cumulative density functions for x are as follows:
CDF(xb) = \int_{-\infty}^{xb} f(x) dx

CDF(xa) = \int_{-\infty}^{xa}- f(x) dx

\int_{xa}^{xb} f(x) dx \mathrel{\widehat{=}} CDF(xb) - CDF(xa) \mathrel{\widehat{=}}  \\ \text{fraction of the probability between xa and xb}

So far as I remember to take the difference between two CDFs of a normal distribution works only if both x values are positive (if x>\mu=0). Is that correct? If yes
how can this criterion be corretly implemented in the equations above (correct notation)?
What's about the equations? Are they correct from a mathematical point of view and is the notation correct in general? How can it be improved?

Thank you