I'm a little unsure about an example of a surface integral I've come across, in which the method of projection is used. The example finds the surface area of a hyperbolic paraboloid given by z=(x2-y2)/2R bounded by a cylindrical surface of radius a, such that x2+y2=<2. The first issue I'm having is in understanding the function of the cylinder - if we project the area element of the surface onto the xy plane then integrate this, surely the projection of the whole paraboloid onto this plane is not in the form of a circle, thus finding the integral of the projected area element would either involve limits too large or small, i.e we would get too much or too little of the surface. If it helps the working is S=∫dxdy√(1+x2/R2+y2/R2) where the latter square root is the projection factor between dS (on the surface) and dA (on the xy plane). S=∫ (from zero to a) 2πrdr√(1+r2/R2). I'm also not sure about what is going on in this step - some kind of change of coordinates maybe? Then let u=r2/R2 and solve for (2∏R2/3)[(1+a2/R2)3/2-1] Thanks for any help in advance.