I gave you a link which shows you how to work out the integrand and do the conversion.
You have to scroll down to the bit where it shows you the example related to your problem.
I suspect you are still thinking in terms of integrating some function provided. Get that out of your head - the dS takes care of that for you. (Either that or you forgot to do the cross product...)
The thinking:
In general - when you have a flat area to calculate, you stamp the area with small squares and count them. dS is that small area. If the rea is flat then the sides of the small squares are dx and dy then the surface area of each is dS=dxdy. The Area of the whole surface is the sum of all the little areas like this:
\int_{Y} \int_{X} dxdy ... where X and Y represent the limits of integration.
Notice how there is apprently no function to integrate? The function in question is g(x,y)=1 - because the surface is flat, it's slope in the x and y directions is 1.
The area below a function in the x-y plane is a special case of this that simplifies to a single integral - so it is easier to teach.
As a double integral, the area under f(x) is:
\int_{a}^{b} \int_{y=0}^{y=f(x)} 1dydxIf the area is not flat, then you project the squares onto the surface - so they look like parallelograms - the sides will be longer depending on the slope of the surface in their directions (hint: partial derivatives) and dS will be in terms of dxdy but modified for the area of a parallelogram. Which is basically what you've just done.
But you can also do this in other coordinate systems.
In polar coords, a small area at position (r,\theta) from the z axis, in a plane perpendicular to the z axis, which covers a distance \Delta r in the r direction, and an angle \Delta \phi in the \phi would be roughly dS=(r\Delta \phi) \Delta r - can you see why? In the limit that \Delta \phi and \Delta r are very small, then this area is exact with:
dS = r dr d\phi
BUT: the surface is not flat!
A parabaloid about z in cylindrical-polar coords would be:
z=a(r+b)^2
So the slope varies in the r direction - making the sides of dS longer in the r direction.
A quick sketch of the situation will help.
(BTW: You'll have noticed how useful LaTeX is by now - it is really worth learning.)