Mr. Delaforce,

The surface area of a spherical cap or spherical dome (i.e a portion of a sphere cut off by a plane) can be found in math books. See for example http://mathworld.wolfram.com/SphericalCap.html

What API says is that DLR is the nominal weight of roof plate plus any attached structural, which is correct in my opinion; such force appears in a free-body diagram of the roof.