You don't need to multiply by g because weight (in pounds), as opposed to mass (in "slugs"), already is the force of gravity on a object. IF you were given mass density instead of weight density, then you would have to multiply by g.
The problem is :
Calculate the work against gravity required to build the structure out of brick, with density 80 lb/ft^3
Structure is a cylindrical tower of height 20 ft and radius 10 ft.
The solution has that the area is 100 pi ft^2 and so the volume per layer is simply 100 pi delta y ft^3
The weight would then be 8000 pi delta y lb. (just multiplying by density)
So the total work against gravity is
integral (0 to 20) 8000 piydy
The only thing I don't understand is why you don't have to multiply by g=32...in most other work problems you do...is it because this says the work done AGAINST gravity?