A.
x[y_]:=y^2/2; dx=D[x[y],y]; Integrate[x[y]*y*Sqrt[1+dx^2],{y,0,2}]
B.
x[y_]:=y^2/2; dx=D[x[y],y]; Integrate[x[y]*y*sqrt[1+dx^2],{y,0,2}]
C.
x[y_]:=y^2/2; dx=D[x[y],y]; Integrate[x[y]*y*Sqrt[1+dx^2],y,0,2]
D.
x[y_]:=y^2/2; dx=D[x[y],y]; Integrate[x[y]*y*Sqrt[1+dx^2],{x,0,2}]