you will have to do three separate integrals, one for each side of the triangle. for each, your r(t) will be the vector function for the line.
Here is how to proceed:
Assuming we are going from (0,0) to (1,0) to (1,2), call the line connecting (0,0) to (1,0) , the line connecting (1,0) and (1,2) and the line connecting (1,2) back to (0,0) .
Then we have .
lets concentrate on .
for , we have the line , with x ranging from 0 to 1. thus, we can parametrize the line by:
, , for
and so, , , and our integral for becomes:
now do the same for and .
(note, there is no need to introduce a here. we could have parametrized our line by , for )