If you want to project the cylinder on the coordinate plane of zx or yz, you can't project it on the xy plane, because the projection of the cylinder on the xy plane is just a curve, not an area.

Here is the projection to the zx plane, considering the symmetry.

The equation of a cylinder is y = √ (a 2-x 2), and the surface element ds = √ [1+(α y/α x) 2+(α y/α z) 2] dzdx = a/√ (a 2-x 2) dzdx, which leads to the above.