Manipulate[
Show[
{
ParametricPlot3D[{s,t (s^2),0},{s,0,1},{t,0,1},PlotStyle->Opacity[a],Mesh->None],
ParametricPlot3D[{s ,t (s^2),t (s^2)},{s,0,1},{t,0,1},PlotStyle->Opacity[b],Mesh->None],
ParametricPlot3D[{s,(s^2),t (s^2)},{s,0,1},{t,0,1},PlotStyle->Opacity[c],Mesh->None],
ParametricPlot3D[{1,s,t s},{s,0,1},{t,0,1},PlotStyle->Opacity[d],Mesh->None]},
AxesLabel->{x,y,z},PlotRange->{{0,1},{0,1},{0,1}},Ticks->{{0,1},{0,1},{0,1}}
],
{{a,1,"bottom"},0,1,.5},
{{b,1,"top"},0,1,.5},
{{c,1,"triangle"},0,1,.5},
{{d,1,"curved wall"},0,1,.5},
ControlPlacement->Left
]

Check

Print["Original: (∫_0^1)(∫_0^(x^2))(∫_0^y)1\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x = ",(∫_0^1)(∫_0^(x^2))(∫_0^y)1\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x];
Print["(a.) (∫_0^1)(∫_(Sqrt[y])^1)(∫_0^y)1\[DifferentialD]z\[DifferentialD]x\[DifferentialD]y = ",(∫_0^1)(∫_(Sqrt[y])^1)(∫_0^y)1\[DifferentialD]z\[DifferentialD]x\[DifferentialD]y ];
Print["(b.) (∫_0^1)(∫_(Sqrt[z])^1)(∫_z^(x^2))1\[DifferentialD]y\[DifferentialD]x\[DifferentialD]z = ",(∫_0^1)(∫_(Sqrt[z])^1)(∫_z^(x^2))1\[DifferentialD]y\[DifferentialD]x\[DifferentialD]z  ];
Print["(c.) (∫_0^1)(∫_0^(x^2))(∫_z^(x^2))1\[DifferentialD]y\[DifferentialD]z\[DifferentialD]x = ",(∫_0^1)(∫_0^(x^2))(∫_z^(x^2))1\[DifferentialD]y\[DifferentialD]z\[DifferentialD]x ];
Print["(d.) (∫_0^1)(∫_0^y)(∫_(Sqrt[y])^1)1\[DifferentialD]x\[DifferentialD]z\[DifferentialD]y = ",(∫_0^1)(∫_0^y)(∫_(Sqrt[y])^1)1\[DifferentialD]x\[DifferentialD]z\[DifferentialD]y ];
Print["(e.) (∫_0^1)(∫_z^1)(∫_(Sqrt[y])^1)1\[DifferentialD]x\[DifferentialD]y\[DifferentialD]z = ",(∫_0^1)(∫_z^1)(∫_(Sqrt[y])^1)1\[DifferentialD]x\[DifferentialD]y\[DifferentialD]z  ];