Plot3D[x y,{x,0,1},{y,0,1}]

?Region*

art=RegionPlot3D[And[y≥0,y≤1,x≥y,x≤2-y,z≥0,z≤x*y],{x,0,2},{y,0,1},{z,0,1},PlotStyle->Directive[Yellow,Opacity[0.5]],Mesh->None,PlotPoints->{200,200,200},PlotRange->{{0,2},{0,1},{0,1}},BoxRatios->{2,1,1}]

arts=RegionPlot3D[And[y≥0,y≤1,x≥y,x≤2-y,z≥0,z≤x*y],{x,0,2},{y,0,1},{z,0,1},PlotStyle->Directive[Yellow,Opacity[0.5]],Mesh->None,PlotPoints->{100,100,100},PlotRange->{{0,2},{0,1},{0,1}},BoxRatios->{2,1,1}]

artsp=Show[ParametricPlot3D[{(1-s)y+s(2-y),y,((1-s)y+s(2-y))*y},{y,0,1},{s,0,1},Mesh->False,PlotStyle->Directive[Yellow,Opacity[0.75]]],ParametricPlot3D[{t,t,(1-s)*0+s (t^2)},{t,0,1},{s,0,1},Mesh->False,PlotStyle->Directive[RGBColor[0.5,0,0],Opacity[0.5]]],ParametricPlot3D[{1+t,1-t,(1-s)*0+s(1- (t^2))},{t,0,1},{s,0,1},Mesh->False,PlotStyle->Directive[RGBColor[0.5,0,0],Opacity[0.5]]]]

yy=.75;Show[artsp,Graphics3D[{Opacity[0.75],Polygon[{{yy,yy,0},{2-yy,yy,0},{2-yy,yy,(2-yy)yy},{yy,yy,(yy^2)},{yy,yy,0}}]}],ImageSize->300]

Manipulate[Show[artsp,Graphics3D[{{Blue,Opacity[0.75],Polygon[{{yy,yy,0},{2-yy,yy,0},{2-yy,yy,(2-yy)yy},{yy,yy,(yy^2)},{yy,yy,0}}]},{Black,Thickness[0.001],Line[
{{yy,yy,0},{2-yy,yy,0},{2-yy,yy,(2-yy)yy},{yy,yy,(yy^2)},{yy,yy,0}}]}}],ImageSize->300],{{yy,0.5},0,1}]

Integrate[Integrate[Integrate[1-z,{z,0,x y}],{x,y,2-y}],{y,0,1}]

N[{(11/45),(2/9)}]