Evaluate line integral of (x+y) ds where c is the straight-line segment x=t, y=(1-t), z=0, from (0,1,0) to (1,0,0)

In order to evaluate the line integral, we have to express everything in terms of the parameter t. Since we have x, y, and z in terms of t already, we specifically need to worry about ds.

The small piece of the curve C associated with the small changes dx, dy, and dz has length . Using this, we can represent the length of ds with the associated change in t as .

What are the limits of integration in terms of t? By drawing the curve and/or plugging in the given points into the equations for x, y, and z in terms of t, we can see that the curve C is traversed by t as it goes from 0 to 1.

Putting all this together and evaluating, we get