Homework, 15.2: #10, 21, 25, 34, and 38, p. 1011
MAT329
Andy Long

#10, p. 1011:

f[x_,y_]:=E^(x+3y)
Plot3D[f[x,y],{x,0,3},{y,0,1}]
Integrate[f[x,y],{x,0,3},{y,0,1}]
N[%]

#21:

f[x_,y_]:=y E^(-x y)
Plot3D[f[x,y],{x,0,2},{y,0,3}]
Integrate[f[x,y],{x,0,2},{y,0,3}]
N[%]

#25:

f[x_,y_]:=(4x+6y+15)/2
Plot3D[f[x,y],{x,-1,2},{y,-1,1}]
Integrate[f[x,y],{x,-1,2},{y,-1,1}]
N[%]
6*f[1/2,0]

#34

RegionPlot3D[2-x^2-y^2>z>E^(-x^2) Cos[x^2+y^2],{x,-1,1},{y,-1,1},{z,-0.2,2},Mesh->False]
(* That's a nice way to graph it! *)
f[x_,y_]=(E^(-(x^2)))*Cos[(x^2)+(y^2)]
g[x_,y_]=2-(x^2)-(y^2)
Plot3D[{f[x,y],g[x,y]},{x,-1,1},{y,-1,1}]
N[(∫_(-1)^1)(∫_(-1)^1)(g[x,y]-f[x,y])\[DifferentialD]x\[DifferentialD]y]
(* what do we do with a complex answer? Throw away the imaginary part!:) *)

#38: Hooray! Use symmetry....

f[x_,y_]:=1+x^2Sin[y]+y^2Sin[x]
Plot3D[f[x,y],{x,-Pi,Pi},{y,-Pi,Pi}]
Integrate[f[x,y],{x,-Pi,Pi},{y,-Pi,Pi}]
N[%]