Homework, Section 15.3
MAT329
Andy Long

#9, p. 1019

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

#14, p. 1020

RegionPlot[y≤3x&&y>=x^2,{x,0,3},{y,0,9},AxesLabel->{x,y}]
f[x_,y_]:=x y
Int=Integrate[f[x,y],{x,0,3},{y,x^2,3x}]
Int=Integrate[f[x,y],{y,0,9},{x,y/3,Sqrt[y]}]

#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]}]

#56

r1=RegionPlot[x≤y-y^3&&x≥-1&& x≥Sqrt[y]-1,{x,-1,1},{y,-1,1},PlotPoints->50];
r2=RegionPlot[x≤y-y^3&&x≥-1,{x,-1,1},{y,-1,0},PlotPoints->50];
Show[r1,r2]
Int1=Integrate[y,{y,-1,0},{x,-1,y-y^3}]
Int2=Integrate[y,{y,0,1},{x,Sqrt[y]-1,y-y^3}]
Int1+Int2