The basic idea is that you use your constraint, i.e. the surface area, to reduce the expression for the volume from 2 variables to 1 variable. Via the constraint you'll have an expression for y in terms of x or vice versa, whichever is more convenient, and you plug that into your volume formula.

That whole mess gives you the volume as the function of a single variable and you know how to maximize that. Just find the zero of the derivative.

Of course you have to come up with formulas for the surface area and volume of your surface of revolution but you can figure that out.

(in case you can't remember you make a surface by spinning a curve around the center 2pi radians. To make a volume you spin the area under a curve around the center 2pi radians.

In the first case the surface area is the length of the curve times 2pi. For the volume it's the area under the curve times 2pi.)