Can someone check my work? The question was: find a parametric representation of the portion of the surface $x+3y-z=5$ with $x\geq0, y\geq0$, and $x^2+y^2\leq 1$.
I answered: $x=\cos\theta$, $y=\sin\theta$ and $z=\cos\theta + 3\sin\theta -5$ with $0\leq\theta\leq \frac{\pi}{2}$.
Thanks in advance.
Based on the wording of your question. No, you are not correct. As I stated in the comments, any point $(\cos \theta, \sin \theta, \cdot )$ lies strictly on the unit circle on the $xy$-plane. What you want is the plane region entrapped in the cylinder $x^2 + y^2 \le 1$.
Here is a plot of both surfaces for $x \ge 0$, and $y \ge 0$:
Here is your parametrized curve $\vec r(\theta) = \langle \cos \theta, \sin \theta, \cos \theta +3 \sin \theta - 5 \rangle$:
It should be clear that your parametrization has only captured the intersection of these two surfaces.
Edit:
A correct parametrization could be:
$$ z = r \cos \theta + 3r \sin \theta - 5$$
where $0 \le r \le 1$ and $0 \le \theta \le \frac{\pi}{2}$ in the cylindrical coordinate system.