The "Total Ion Current Area" is a single number representing the integral over time of the TIC chromatogram.
The way that Skyline calculates the areas is by adding up the areas of rectangles.
Here's what our code for calculating that integral looks like:
https://github.com/ProteoWizard/pwiz/blob/3456c118dbef6587c7dc952ea6bd84a0342527db/pwiz_tools/Skyline/Model/Results/TimeIntensities.cs#L299

If you wanted to calculate the area under the TIC chromatogram yourself, you would need to make sure that you are multiplying each intensity by the difference between its retention time and its neighbors. If you were to simply sum all of the intensities in the chromatogram you would get a very different number.
Are you getting a different number when you calculate the area under the TIC chromatogram?

By the way, Skyline usually gets the data for the TIC chromatogram by reading the TIC chromatogram directly from the raw file, and using the subset of those points which correspond to MS1 spectra. There are some cases where Skyline cannot use that chromatogram, in which case Skyline sums the intensities on all of the MS1 spectra itself.

If you still have questions about what Skyline is doing you can send us your Skyline document and some of your .raw files.
In Skyline you can use the menu item:
File > Share
to create a .zip file containing your Skyline document and supporting files including extracted chromatograms. The "Share Skyline Document" dialog also gives you the option to include raw files in .zip.

Files which are less than 50MB can be attached to this support request. You can upload larger files here:
https://skyline.ms/files.url
-- Nick