Example  1

Plot3D[x^2*y,{x,0,1},{y,0,1},RegionFunction->Function[{x,y,z},x^2<y<Sqrt[x]],AxesLabel->{x,y,z}]

Example 2

Plot3D[Cos[x]*Cos[y],{x,0,2Pi},{y,0,Pi},RegionFunction->Function[{x,y,z},y<x<2Pi-y],AxesLabel->{x,y,z}]

Example 3 (#53)

Plot3D[Exp[-(x^2+y^2)^2],{x,0,1/2},{y,0,1/2},RegionFunction->Function[{x,y,z},x^2+y^2<1/4],AxesLabel->{x,y,z}]

NIntegrate[Exp[-(x^2+y^2)^2],{x,0,1/2},{y,0,Sqrt[1/4-x^2]}]