I need to know the formula for calculating area's under a curve of normal distributions
The area under the standard normal curve, from x= a to x= b, is, as I suspect you knew,
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.