I have to evaluate the surface area of the cylinder $$x^2+y^2=4$$ between the planes $z=0$ and $x+y+z=2$. I was trying to use the following formula to area $$\iint_{D}|\sigma_{\theta}\times\sigma_{z}|dA$$ where $\sigma(\theta,z)$ is its parameterization.
My doubt is: may I write $\sigma$ as follows? $$\sigma(\theta,z)=(2\cos\theta,2\sin\theta,z)$$ with $0\leq\theta\leq2\pi$ and $0\leq z\leq2-2\cos\theta-2\sin\theta$.