Problem 1

Plot3D[6x y^2,{x,0,1},{y,0,1},AxesLabel->Automatic,ColorFunction->"TemperatureMap",BoxRatios->{1,1,1},PlotRange->All,FaceGrids->{{0,0,-1}}]

Integrate[6x*(y^2),{y,0,1},{x,0,1}]

Integrate[6x*(y^2),{y,0,1}]

Integrate[(x+y)6x*(y^2),{x,0,1},{y,0,1}]

N[17/12]

Integrate[(x*y)6x*(y^2),{x,0,1},{y,0,1}]

Problem 2

Plot3D[{0,3x+3y},{x,0,1},{y,0,1-x},AxesLabel->Automatic,ColorFunction->"TemperatureMap",BoxRatios->{1,1,1},FaceGrids->{{0,0,-1}}]

Confirm the volume is 1:

Integrate[3x+3y,{x,0,1},{y,0,1-x}]

Integrals for part (c):

E(X+Y)

Integrate[(x+y)(3x+3y),{x,0,1},{y,0,1-x}]

E(X Y)

Integrate[x y(3x+3y),{x,0,1},{y,0,1-x}]

E(X)

Integrate[3x+3y,{y,0,1-x}]//Simplify

Integrate[x((3/2))(1-(x^2)),{x,0,1}]

E(Y)

Integrate[3x+3y,{x,0,1-y}]//Simplify

Integrate[y((3/2))(1-(y^2)),{y,0,1}]