I'm trying to find the area of parametric surface of a cylinder projected onto a plane. The parametric equation of a cylinder is $$S(u, v) = (r\cos(u), r\sin(u), v)$$ the equation for projecting a surface onto a plane with unitized normal vector $\overrightarrow{N}$ is: $$S_n(u,v) = S(u,v) - \langle S(u,v),\,\overrightarrow{N}\rangle *\overrightarrow{N} $$ And finally, for calculating the area of the projection we should do the following: $$ S_u = \frac{dS}{du}; \;\;\;\;\;\;\;\;\;\;\;S_v = \frac{dS}{dv}; $$ $$ A = \iint_D\|S_u\times S_v\|\,du\,dv $$ But this integral is way too hard to solve like this in cartesian coordinates for arbitrary $D$ regions, I want to make a change of variables to cylindrical coordinates, such that $S(u,v)$ becomes $$ S_{cylinder}=(r, u, v) $$ Making the integral easier to solve, but I cant seem to find how to correctly change the integral such that the resulting area matches the real value. The usual change of coordinates method uses a Jacobian to adjust the differential terms, but in this case wouldn't the differential terms stay the same? Is it even possible to do this change of variables in this situation?
For testing I've been trying to project the half cylinder $S(u,v)$ with $-\frac{\pi}{2}\leq u\leq\frac{\pi}{2}$ and $0\leq v\leq1$, onto the plane with normal vector $\overrightarrow{N} = (1, 0, 0)$ in cartesian coordinates, which has the same values for its cylindrical coordinates. The resulting area should be $2r$.