$F=(z-y)i+(x-z)j+(2y-x)k$
Let $C$ be a curve formed by an intersection of the plane $2x-z=0$ with the cylinder of elliptical cross section $x^2+(y^2)/9=1$, assuming $y$ is parametrized along $C$ via $y=3\sin{t}$, where $t\in[-\frac{\pi}{3},\frac{\pi}{3}]$.
Find the line integral $\int_C{F \cdot d\vec{r}}$ over $C$ (assume corresponding parametric equation for $x$ is $x>0$)
I have parametrized it $x=\cos{t}$, $y=3\sin{t}$ and $z=2\cos{t}$, but I can't seem to work out what the parametrized curve is i.e. the line $r$.
Hint: your integral can be written as follows:
$$I = \int_C F(\mathbf{x}) \cdot \mathrm d \mathbf{x},$$ where $\mathbf{x} = (x,y,z)$ and therefore $\mathrm{d} \mathbf{x}$ is the (tangent) differential vector of the curve given by $C$. Thus, we can write, since $\mathrm{d} \mathbf{x} = \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} \, \mathrm{d}t$:
$$I = \int_{t_0}^{t_F} F(\mathbf{x}(t)) \cdot \mathbf{x}'(t) \, \mathrm{d}t,$$ where $t_0 = -\pi/3$ and $t_F = \pi/3$. Substitute back your data and perform the 1D integral. I'm sure you can take it from here.
Cheers!