Determining the percentage area of the moon's disk shaded with respect to time I've looked online and I've found sources such as this paper on a lunar phase simulator as well as this article about lunar phases.
I'd like to be able to model the area of the lunar disk which is shaded with respect to the moon's angle or time of the lunar month and I'm not sure how to go about doing this. In essence, I would like to be able to find a function f(x) where I can feed in a value such as 0 (meaning 0 days since the start of the lunar month) and I'd be able to get an output like 0 since it's a new moon. Likewise, f(14ish) should be new moon or fig 5 on the image below and so I'd get a value close to 1.
I'm focused a bit more on the maths rather than real values, so I'm willing to make assumptions such as the moon and earth being perfect spheres for the sake of the model.

I know there's a lot of maths involved and I'm going to have to do some integration, but I'd appreciate it if someone could point me to sources to get started on this since the papers I'm finding are either children's lunar phase worksheets or not related to this topic.
If I interpret your question correctly, you want to calculate the area of the moon disk as seen from Earth which is shaded, correct? You might then be looking for a way to calculate the position of the moon's terminator line. For this, you might have to dive into celestrial mechanics, more specifically about the ephemerides of moon and Earth. This is necessary as you  have to calculate the position of Earth and moon in an appropriate sun-centered coordinate system.
In a basic course on celestial mechanics this does not necessarily occur since the priority there is to calculate the stability of the orbit of objects.
Related questions and answers

Get a table of Moon phase angle per day
How to calculate the moon phase angle?

References

Klaus Hempe, Jürgen Molt: Sterne im Computer. Berechnungsprogramme für den Hobby- Astronomen (1999). IMHO an accessible explanation of the numerics, but in German.

http://www.internetsv.info/MoonCalc.html seems helpful.

Wolframalpha's moon phase calculation always give you the illuminated percentage of the moon disk.