Integrate[Exp[((x-y)/(x+y))],x]

Integrate[Exp[((x-y)/(x+y))],{x,0,1-y},Assumptions->And[y>0,y<1]]

Integrate[Integrate[Exp[((x-y)/(x+y))],{x,0,1-y},Assumptions->And[y>0,y<1]],{y,0,1}]

This is the plot of the funcion.

Plot3D[Exp[((x-y)/(x+y))],{x,0,1},{y,0,1},AxesLabel->{"x","y","z"},BoxRatios->{1,1,1}]

In class we did not calculate the volume of the house with the foundation on the unit square.  Its volume is

Integrate[Integrate[Exp[((x-y)/(x+y))],{x,0,1},Assumptions->And[y>0,y<1]],{y,0,1}]

Since this formula involves functions that we do not learn in calculus, the above volume is not calculable by what is learned in calculus.

However, in our problem we calculated the volume of the house whose foundation is the triangle with verteces (0,0), (1,0) and (0,1). To picture that house I have to use ParametricPlot3D.

ParametricPlot3D[{t x,t(1-x),Exp[((t x-t(1-x))/t)]},{x,0,1},{t,0.01,1},AxesLabel->{"x","y","z"},BoxRatios->{1,1,1}]

As we have seen in class, the volume of the above house is relatively easy to calculate using the change of variables in double integrals.

Let us now try to produce a nicer picture of this house.

VP={1.3`,-2.4`,2.`}

Show[ParametricPlot3D[{t x,t(1-x),Exp[((t x-t(1-x))/t)]},{x,0,1},{t,0,1},Mesh->{40,50}],Graphics3D[{{
Opacity[1],Polygon[{{0,0,0},{1,0,0},{1,0,E},{0,0,E}}],
Polygon[{{0,0,0},{0,1,0},{0,1,1/E},{0,0,1/E}}]
},
{
Opacity[1],Polygon[Join[{{1,0,0},{0,1,0}},Table[{x,1-x,Exp[((2 x-1)/1)]},{x,0,1,0.01}]]]}
}],AxesLabel->{"x","y","z"},PlotRange->{{0,1},{0,1},{0,E}},BoxRatios->{1,1,1},ImageSize->250,ViewPoint->{-3.1693710277036926`,1.017061522758984`,0.6088293255713182`}]

Show[ParametricPlot3D[{t x,t(1-x),Exp[((t x-t(1-x))/t)]},{x,0,1},{t,0,1},Mesh->{40,50}],Graphics3D[{{
Opacity[1],Polygon[{{0,0,0},{1,0,0},{1,0,E},{0,0,E}}],
Polygon[{{0,0,0},{0,1,0},{0,1,1/E},{0,0,1/E}}]
},
{
Opacity[1],Polygon[Join[{{1,0,0},{0,1,0}},Table[{x,1-x,Exp[((2 x-1)/1)]},{x,0,1,0.01}]]]}
}],AxesLabel->{"x","y","z"},PlotRange->{{0,1},{0,1},{0,E}},BoxRatios->{1,1,1},ImageSize->250,ViewPoint->{-0.9897680706625`,3.1306045633112745`,0.8183362600249744`}]

Show[ParametricPlot3D[{t x,t(1-x),Exp[((t x-t(1-x))/t)]},{x,0,1},{t,0,1},Mesh->{40,50}],Graphics3D[{{
Opacity[0.85],Polygon[{{0,0,0},{1,0,0},{1,0,E},{0,0,E}}],
Polygon[{{0,0,0},{0,1,0},{0,1,1/E},{0,0,1/E}}]
},
{
Opacity[1],Polygon[Join[{{1,0,0},{0,1,0}},Table[{x,1-x,Exp[((2 x-1)/1)]},{x,0,1,0.01}]]]}
}],AxesLabel->{"x","y","z"},PlotRange->{{0,1},{0,1},{0,E}},BoxRatios->{1,1,1},ImageSize->250,ViewPoint->{-3.162716978927727`,-1.1967256701508124`,0.12275578032978565`}]