Notice that the work needed to complete the integrals over the entire domain above was pretty serious. To determine probabilities for a given interval is however not possible in general and therefore approximations are needed. When using TI graphing calculators, you can use
\begin{equation*}
P( a \lt x \lt b ) = \text{normalcdf}(a,b,\mu, \sigma).
\end{equation*}
Or you can use the calculator below.