The area under the standard normal curve, from x= a to x= b, is, as I suspect you knew,

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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 isdefinedas that integral). That iswhythere are tables of the standard normal distribution, which are themselves developed by numerical integration.