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 [tex]ds = \sqrt{dx^2 + dy^2 + dz^2}[/tex]. Using this, we can represent the length of ds with the associated change in t as [tex]ds = \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2 } \ dt[/tex].
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
[tex] \int\limits_C {x(t) + y(t)} \, ds [/tex]
[tex]= \int\limits_C {(x(t) + y(t)) \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2} \, dt [/tex]
[tex]= \int\limits^1_0 {(t + (1 - t)) \sqrt{(1)^2 + ({-1})^2 + (0)^2}} \, dt[/tex]
[tex]= \int\limits^1_0 {\sqrt{2}} \, dt [/tex]
[tex]= \bf \sqrt{2}[/tex]
