Calculating boundaries for a multiple integral from inequalities

It is right?

reg = ImplicitRegion[ y <=  Sqrt[x] && z == 1 + x + y &&  y >= 0 && x <= 1, {x, y, z}]
% // Region
Integrate[6*x*y, {x, y, z} ∈ reg]

Sqrt[3]

Another way

According to the definition of the integral along a surface.

The normal of the surface z-(1+x+y)==0 is Grad[z - (1 + x + y), {x, y, z}

Integrate[
 6 x*y*Norm[Grad[z - (1 + x + y), {x, y, z}]], {x, y} ∈ 
  ImplicitRegion[y <= Sqrt[x] && y >= 0 && x <= 1, {x, y}]]

Sqrt[3]