# Whats the equation to calculate the area under the curve of a normal distribution, given an upper and lower standard deviation?

Lets say I want to find out the area under the graph of normal distribution curve, between X1=standard deviation of -0.5 and X2 = standard deviation of 0.5. Is there a formula for this?

Case study: find the percentage chance of a stock remaining within +0.5 and -0.5 standard deviations within one trading day, given the daily implied volatility of that stock.

This is simply the integral of the pdf from -0.5 to 0.5 (scaled to the SD of the distribution), also known as the cumulative distribution function or cdf. The cdf(x) function is indicated on the following wikipedia link: Normal Distribution.
The normal cdf(x) function computes the integral on [-Infinity, x], so to compute on your interval [x1,x2], is simply cdf(x2) - cdf(x1).
So if you meant [-.5, +.5] SDs,then would evaluate cdf(0.5*sd) - cdf(-0.5*sd).