Examples, Section 15.3
MAT329
Andy Long

#1, p. 1019

RegionPlot[0<x<Sqrt[y],{x,0,2},{y,0,4}]
f[x_,y_]:=x y^2
(* order matters in mathematica, and goes from outside in! *)
Integrate[f[x,y],{y,0,4},{x,0,Sqrt[y]}]
(* Whoops.... *)
Integrate[f[x,y],{x,0,Sqrt[y]},{y,0,4}]

#6, p. 1019

RegionPlot[0<w<E^v,{v,0,1},{w,0,E}]
f[v_,w_]:=Sqrt[1+E^v]
Integrate[f[v,w],{v,0,1},{w,0,E^v}]

#15, p. 1019

RegionPlot[x>y^2&&y>x-2,{x,0,4},{y,-1,2}]
Solve[y^2==y+2,y,Reals]
f[x_,y_]:=y
Integrate[f[x,y],{y,-1,2},{x,y^2,y+2}]