Integrating a Line and Distance

If I have a line from point 0,0 to 10,10, the length is Rad(200). My question is, if I first go from 0,0 to 10,10 by going from 0,0 to 10,0 - then from 10,0 to 10,10 - the distance is 20. If I half each of those, so I'm moving 0,0 - 0,5 - 5,5 - 5,10 - 10,10 - the distance is still 20. Let's say I half the distance of each section again so its 0,0 - 0,2.5 - 2.5,2.5, etc. you get the idea - the distance is always 20. Eventually when I make the distances short enough, it approaches my first straight line. As the number of segments approach infinity, it should become that straight line. But 20 doesn't equal Rad(200). Why does this work with area but not distances? I know that this doesn't work because there's always a tiny margin of error, and when it's multiplied by the near infinity number, you get the difference between 20 and Rad(200.) But this doesn't happen in Area Integration problems.... Why not?