We have to find the path integral of the following vector along the given curve
$F(x,y,z)=(x^2,z,y)$ along the curve $\gamma$ that parameterizes the line
{$L={(x,y,z) \in R^3: x^2+y^2=1, z=2y+1,x\geq 0}$} so that its projection on the z = 0 plane is counterclockwise oriented.
I found this in the solved exercise session in my university practice session and the professor immediately says that the parametrization of the line $L$ is:
$\gamma(t)=(\cos(t),\sin(t),\sin(t)+1), t\in[-\frac {\pi} 2,\frac \pi 2]$
If possible can you help me understand HOW he parametrized the line L?
I know the rest of the way is just using the formula $\int ^{2\pi}_0F(\gamma(t)\gamma'(t)dt)$ and from here on is easy . But im more interested on how he parametrized the line
$x^2+y^2=1$ is a unit circle. It can be parameterized as $(x,y) = (\cos(t),\sin(t)),t = [-\pi,\pi]$. This is counter clockwise.
$z = 2y+1$ so $z = 2\sin(t)+1$.
$\gamma(t)=(\cos(t),\sin(t),2\sin(t)+1)$
$x \ge 0$ occurs when $\cos(t) \ge 0$, $-\frac{\pi}{2} \le t \le \frac{\pi}{2}$.