Homework, Section 15.3
MAT329, Fall, 2015
Andy Long

#9, p. 1019

f[x_,y_]:=x
Int=Integrate[f[x,y],{x,0,Pi},{y,0,Sin[x]}]

#28, p. 1020

RegionPlot3D[z<=x&&y>=x&&y<=2-x,{x,0,1},{y,0,2},{z,0,1},PlotPoints->50,AxesLabel->{x,y,z}]
f[x_,y_]:=x
Int=Integrate[f[x,y],{x,0,1},{y,x,2-x}]

#48, p. 1020:

RegionPlot[y≥ArcTan[x]&&y≤Pi/4,{x,0,1},{y,0,Pi/4},PlotPoints->50]
Int=Integrate[f[x,y],{x,0,1},{y,ArcTan[x],Pi/4}]
Int=Integrate[f[x,y],{y,0,Pi/4},{x,0,Tan[y]}]

Quiz answers....

x=.5
RegionPlot[z<1-y&&y>Sqrt[x],{y,0,1},{z,0,1},PlotPoints->50]
y=.5
RegionPlot[z<1-y&&x<y^2,{x,0,1},{z,0,1},PlotPoints->50]
z=.5
RegionPlot[x<1-z^2&&Sqrt[x]<y<1-z,{x,0,1},{y,0,1},PlotPoints->50]
Clear[x,y,z]

Int=Integrate[x y^2z^3,{x,0,1},{y,Sqrt[x],1},{z,0,1-y}]
Int=Integrate[x y^2z^3,{x,0,1},{z,0,1-Sqrt[x]},{y,Sqrt[x],1-z}]
Int=Integrate[x y^2z^3,{y,0,1},{x,0,y^2},{z,0,1-y}]
Int=Integrate[x y^2z^3,{y,0,1},{z,0,1-y},{x,0,y^2}]
Int=Integrate[x y^2z^3,{z,0,1},{y,0,1-z},{x,0,y^2}]
Int=Integrate[x y^2z^3,{z,0,1},{x,0,(1-z)^2},{y,Sqrt[x],1-z}]

Int=Integrate[x y^2z^3,{y,0,1},{x,y^2,1},{z,0,1+x}]
Int=Integrate[x y^2z^3,{y,0,1},{x,y^2,1},{z,0,1-y}]
Int=Integrate[x y^2z^3,{y,0,1},{x,0,y^2},{z,0,1-Sqrt[x]}]