(∫_0^1)(1+x y)\[DifferentialD]y

(∫_0^1)%\[DifferentialD]x

(∫_0^1)(∫_0^1)(1+x y)\[DifferentialD]y\[DifferentialD]x

Plot3D[{0,1+x y},{x,0,1},{y,0,1},Mesh->False,Filling->Bottom,PlotStyle->{{LightBlue,Opacity[0.2]},{LightGray,Opacity[0.2]}},BoundaryStyle->{{Thick,Blue},{Thick,Gray}}]

sombra=Plot[{(x^2)-2x,x},{x,0,3},GridLines->Automatic]

picanha=Plot3D[{0,1},{x,0,3},{y,-1,3},Mesh->False,Filling->Bottom,PlotStyle->{{LightBlue,Opacity[0.2]},{LightGray,Opacity[0.2]}},RegionFunction->Function[{x,y,z},(x^2)-2 x≤y≤x],BoundaryStyle->{{Thick,Blue},{Thick,Gray}}]

fatia=ParametricPlot3D[(1-t){1.5,(((1.5))^2)-2(1.5),0}+t{1.5,1.5,0}+{0,0,z},{t,0,1},{z,0,1},Mesh->False,PlotStyle->{LightRed,Opacity[0.5]},BoundaryStyle->{Thick,Red}]

Show[picanha,fatia]

Show[sombra,Graphics[{PointSize[Large],Red,Point[{{1.5,(((1.5))^2)-2(1.5)},{1.5,1.5}}],Line[{{1.5,(((1.5))^2)-2(1.5)},{1.5,1.5}}]}]]

line[x_]:=Graphics[{PointSize[Large],Red,Point[{{x,(x^2)-2x},{x,x}}],Line[{{x,(x^2)-2x},{x,x}}]}];

Manipulate[Show[sombra,line[x]],{x,0,3}]

(∫_((x^2)- 2x)^x)1\[DifferentialD]y

(∫_0^3)%\[DifferentialD]x

(∫_0^3)(∫_((x^2)- 2x)^x)1\[DifferentialD]y\[DifferentialD]x