Given $\vec f(x,y,z)=\left(ye^{xy},\:xe^{xy},\:z+y\right)$ find, through a double integral, the area enclosed by the projection in the $xz$-plane of $C:\vec\gamma(t)=\left(2\cos{(t)},\:2\cos{(t)}+3\sin{(t)},\:-3\sin{(t)}\right),\;t\in[0,2\pi]$.
I know I have to calculate
$$\iint\limits_{P_{xz}(C)}{\text{something}}\:\text dx\text dz$$ but I can not go further except that I did
$$C:\begin{cases}x&=2\cos{(t)}\\y&=2\cos{(t)}+3\sin{(t)}\\z&=-3\sin{(t)}\end{cases}\quad\equiv\quad \begin{cases}x&=2\cos{(t)}\\y&=x-z\\z&=-3\sin{(t)}.\end{cases}$$
How should I finish? Should I clarify that the field is $\mathcal C^1(D)$?
Thank you!