It is not true I believe. It should be a double integral since you have .
Take it from here, I think all should go well.
As Danny says, you should write the equation of the sphere as . To find the region over which to integrate, you need to know where the sphere meets the cone, namely where or in other words . So the integral will be over the region , and the surface area is given by the integral . This cries out for conversion to polar coordinates, and it then becomes . From there on, it's easy.
I think it would be easier to do the all thing in spherical coordinates.
Notice that if the height of the cone is z (a generic point z), and the radius of the base is , so the generating angle for the cone, , is given by
as , and
this makes the integral for the area
it seems much more simpler to me, and no need to change the coordinates in between. the R being 2 or 5 makes no difference you just replace it in the integral as a constant.