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$\displaystyle (\mu = 0, \sigma^{2}=1)$
is $\displaystyle f(x) $

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

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

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

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 $\displaystyle x>\mu=0$). 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