Hi;
I need to know the formula for calculating area's under a curve of normal distributions
without tables.
Thanks.
The area under the standard normal curve, from x= a to x= b, is, as I suspect you knew,
$\displaystyle \frac{1}{\sqrt{2\pi}}\int_a^b e^{\frac{-x^2}{2}}dx$.
If you are asking for a way to find an anti-derivative, and then evaluate at a and b, there is no such way. The anti-derivative cannot be expressed in terms of elementary functions (it can, of course, be expressed in terms of the "Error Function", erf(x), which is defined as that integral). That is why there are tables of the standard normal distribution, which are themselves developed by numerical integration.
You can use a CAS calculator to evaluate the area under a normal curve, they are programmed to be able to do numerical integration and so are slightly more accurate than the tables. But HallsofIvy is correct, there is no known way to evaluate a Gaussian-type definite integral, EXCEPT between $\displaystyle \displaystyle \begin{align*} -\infty \end{align*}$ and $\displaystyle \displaystyle \begin{align*} \infty \end{align*}$, or between 0 and one of the infinities (as it's exactly half the region).