Typos abound in the above post, sorry about that. It was supposed to be:
## \vec R = <t, t, t>, \frac{d\vec R}{dt} = <1,1,1> ##
So if the path is from (0,0,0) to (X,Y,Z), if I were to do it by integrating with dt, would I need to use bounds on t from 0 to, say, T?