It would be useful if that page had some diagrams.
Averaging the cartesian coordinates is what we used school - perhaps it was with smaller polygons.
Seeing what they've written about the "rectangle and trapezium", I think I see what they are getting at. You might have lots of points at one end of the shape. A simple average will then bring the calculated centre over to that side - when it shouldn't.
Yes integration is the way to go. It is a long time since I did this, but it looks/sounds like the momentum calculations we used to do at Univ...
(but in 2d and without the velocity component)