Finding the parameterization of a curve for a line integral problem I have to calculate the work of a particle that travel along a curve, given the following vector field:
$F(x, y, z) = (2z-1, 0, 2y)$
and where the curve is the intersection between:
$s1: z = x^2 + y^2$ and $s2: 4x^2 + 4y^2 + 1 = 4x + 4y$
using the definition of line integrals.
What complicates me of this exercise is parameterize the curve, any help?

There are many ways to parametrize a given curve, but I'll discuss one example here. The equation $4x^2 + 4y^2 + 1 = 4x + 4y$ can be rearranged:
\begin{align*}
4x^2 - 4x + 1 = -4y^2 + 4y -1 + 1 
&\iff (2x-1)^2 = -(2y-1)^2 + 1 \\
&\iff (2x-1)^2 + (2y-1)^2 = 1.
\end{align*}
Knowing this, let us set $2x - 1 = \cos(t)$ and $2y-1 = \sin(t)$, where $t$ ranges from $0$ to $2\pi$. Using the equation $z = x^2 + y^2$, we can also find the parametrization of $z$:
$$
z = \left( \frac{\cos(t) + 1}{2} \right)^2 + \left( \frac{\sin(t) + 1}{2} \right)^2 = \frac{\sin(t)}{2} + \frac{\cos(t)}{2} + \frac{3}{4}.
$$
