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I want to calculate the area under a curve between 250nm to 300nm to find the irradiance in [W/m2], to do this I think must find the function of Spectral Irradiance (sunlight out the atmosphere, yellow function) and then apply integral between the two points, in that case, how is this function? If that is not the case, how calculate the irradiance between 250nm to 300nm?

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The easiest way is the old-fashioned way, using the raw data for spectral irradiance at each wavelength and adding it up. The NREL website has spreadsheets of the raw data (I believe for both AM0 and AM1.5) you can use.

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