How to find a smooth parametrization of a Curve In order to solve a line integral, I need to establish a smooth parametrization of the curve over which it is supposed to be integrated. 
The curve, $D$, is the intersection of the surfaces $x^2 + y^2 = 1$  and $z=x^2$.
To me, a logical parametrization is: $$r(t)=(t, (1-t^2)^{1/2}, t^2)$$
because if $x = t$, then
 $$y = (1-t^2)^{1/2},$$ and $$z = t^2$$
Is this right? I appreciate any help.

No, your parametrization only gives you positive values of $y$.  Since $x^2 + y^2 = 1$ is a circle in the $x-y$ plane, you might try $x = \cos(t)$, $y = \sin(t)$.
