Green's theorem Section properties

[LR11]

I've come across a paper that uses Green's theorem to calculate section properties of any shape that contains lines and circular arcs. Entitled "The State-of-the-Art of Section Property Calculation of Structural Members with Arbitrary Shape" by K.F. Chung, it contains functions (in Appendix C), that are used to calculate the area, first moment of area and second moment of area. I'm having issues in getting the correct answer for first and second moment of area using the functions provided. To itemize the anomalies:
- First moment of area for arcs is missing a multiplier I think.
- Second moment of area for straight lines is missing multipliers I think.
- * Second moment of area for arcs - no idea, need help with this one.

I need help with the item marked with a "*".

Incidentally what method is used by software to calculate section properties?