ClearAll["Global`*"]

myAxis[label_,color_,component_,lo_,hi_]:=Module[{e,g,o},
o={0,0,0};e=o;e[[component]]=1;
g={color,Text[label,1.05 hi e],Arrowheads[0.02],Arrow[{o,hi e}]};
If[lo<0,g=Join[g,{Dashed,Line[{lo e,o}],Dashing[{}]}]];
g];
myPoint[p_,color_,label_]:={color,Text[label,1.1 p],Dashed,Line[{{p[[1]],0,0},{p[[1]],p[[2]],0},p}],PointSize[Medium],Point[p],Dashing[{}]};

xy=ParametricPlot3D[{{x,y,1},{x,y,0}},{x,0,1},{y,0,1},Mesh->1,PlotStyle->Directive[Darker[Yellow],Opacity[0.05]]
];
yz=ParametricPlot3D[{{1,y,z},{0,y,z}},{y,0,1},{z,0,1},Mesh->1,PlotStyle->Directive[Orange,Opacity[0.4]]
];
zx=ParametricPlot3D[{{x,1,z},{x,0,z}},{x,0,1},{z,0,1},Mesh->1,PlotStyle->Directive[Green,Opacity[0.05]]
];
Show[xy,yz,zx,ViewPoint->{100,50,20}];

ax=Graphics3D[
Join[
myAxis["x",Orange,1,-.2,1.5],myAxis["y",Darker[Yellow],2,-0.2,1.5],myAxis["z",Green,3,-.2,1.5],
{
Orange,Arrow[{{1,0.5,0.5},{1.3,0.5,0.5}}],Arrow[{{0,0.5,0.5},{-.3,0.5,0.5}}],
Darker[Yellow],Arrow[{{0.5,1,0.5},{0.5,1.3,0.5}}],Arrow[{{0.5,0,0.5},{0.5,-.3,0.5}}],
Green,Arrow[{{0.5,0.5,1},{0.5,0.5,1.3}}],Arrow[{{0.5,0.5,0},{0.5,0.5,-0.3}}]
}
],Axes->True,ViewPoint->{100,50,20}
];

Here’s a picture of the box, with surface normals.  Orange face normals point in the +(OverHat[i])and -(OverHat[i])

Show[ax,xy,yz,zx, ViewPoint->{100,50,20}]

...and with a vector plot of the vector field (OverVector[F])=(x^2)(OverHat[i]).  The only face contributing to the surface is the one at x=1.  On this face (OverVector[F])is non-zero, and points parallel to the surface normal.

vp=VectorPlot3D[{x^2,0,0},{x,0,1},{y,0,1},{z,0,1}];

Show[ax,xy,yz,zx,vp, ViewPoint->{100,50,20}]

The surface integral is

({{({{∫(OverVector[F])·d(OverVector[S])=(∫_0^1)(∫_0^1)(OverVector[F])(1,y,z)·(OverVector[i])dydz}})= (∫_0^1)(∫_0^1)(1^2)(OverVector[i])·(OverVector[i])dydz=(∫_0^1)(∫_0^1)1dydz=(∫_0^1)((Subscript[(y), 0])^1)dz=(∫_0^1)1dz=1}})