Esercizio 6

Integrazione su rettangoli [a,b]x[c,d]

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

Integrate[(x^3) (y^2)-4 (x^2) y,{x,0,1},{y,0,3}]

Integrate[Sin[x+3 y],{x,0,Pi},{y,0,3 Pi}]

Integrate[y Cos[x y],{x,-(Pi/4),(Pi/4)},{y,0,1}]

Integrate[(E^(2 x+y)),{x,2,4},{y,0,1}]

FullSimplify[(1/2) (((-1+E))^2) (E^4) (1+E+(E^2)+(E^3))]

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

Integrate[(Sqrt[x+y]),{x,0,2},{y,0,2}]

Integrate[x^2+y^2,{x,-1,1},{y,-2,2}]

Manipulate[If[XorY=="y=c",Grid[{{Plot[h/.y->c, {x,-4,4},PlotRange->{-6,6}, AxesLabel->{"x", "z"},PlotStyle->Thick, ImageSize->{275,275}, Filling->Axis],Plot3D[h,{x,-2.1,2.1},{y,-2.1,2.1},PlotRange->6,Mesh->{{-2.1},{{c,Thick}}},AxesLabel->Automatic, ColorFunction->"DarkRainbow",ImageSize->{275,275}]}}],Grid[{{Plot[h/.x->c, {y,-3 Pi,3 Pi},PlotRange->{-6,6}, AxesLabel->{"y", "z"},PlotStyle->Thick,ImageSize->{275,275} , Filling->Axis],Plot3D[h,{x,-2.1,2.1},{y,-2.1,2.1},PlotRange->6,Mesh->{{{c,Thick}},{-2.1}},AxesLabel->Automatic, ColorFunction->"DarkRainbow",ImageSize->{275,275}]}}]],{{h,x y (E^y) ,"f(x,y)"},{x y (E^y) ,(x^3) (y^2)-4 (x^2) y,Sin[x+3 y],y Cos[x y], (E^(2 x+y)), ((1+(x^2))/(1+(y^2))),(Sqrt[x+y]),x^2+y^2}, ControlType->PopupMenu},
{{XorY, "x=c","direction"}, {"x=c","y=c"},ControlType->Setter},{{c,0,"c"},-2,2, Appearance->"Labeled"}]

Esercizio 8

Integrazione su domini semplici

RegionPlot[{0≤y≤(x^2)},{x,0,1},{y,0,1}]
Integrate[x Cos[y] Boole[0≤y≤(x^2)],{x,0,1},{y,0,1}]
Plot3D[x Cos[y],{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},0≤y≤(x^2)]]

RegionPlot[{0≤x≤y},{x,0,1},{y,0,1}]
Integrate[x (Sqrt[(y^2)-(x^2)]) Boole[0≤x≤y],{x,0,1},{y,0,1}]
Plot3D[x (Sqrt[(y^2)-(x^2)]),{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},0≤x≤y]]

RegionPlot[{0≤y≤x},{x,0,1},{y,0,1}]
Integrate[(E^(x^2)) Boole[0≤y≤x],{x,0,1},{y,0,1}]
Plot3D[(E^(x^2)),{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},0≤y≤x]]

RegionPlot[{(x^2)≤y≤(Sqrt[x])},{x,0,1},{y,0,1}]
Integrate[(x+y)  Boole[(x^2)≤y≤(Sqrt[x])],{x,0,1},{y,0,1}]
Plot3D[x+y,{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},(x^2)≤y≤(Sqrt[x])]]

RegionPlot[{0≤y≤x &&0≤y≤-x+4 },{x,0,4},{y,0,2}]
Integrate[x (E^y) Boole[0≤y≤x &&0≤y≤-x+4],{x,0,4},{y,0,2}]
Plot3D[x (E^y),{x,0,4},{y,0,2},Mesh->None,RegionFunction->Function[{x,y},0≤y≤x &&0≤y≤-x+4]]

RegionPlot[{-x+2≤y≤1+(Sqrt[1-(x^2)])},{x,0,1},{y,1,2}]
Integrate[x  Boole[-x+2≤y≤1+(Sqrt[1-(x^2)]) ],{x,0,1},{y,1,2}]
Plot3D[x ,{x,0,1},{y,1,2},Mesh->None,RegionFunction->Function[{x,y},-x+2≤y≤1+(Sqrt[1-(x^2)]) ]]

RegionPlot[{0≤x≤(y^2)},{x,0,1},{y,0,1}]
Integrate[(E^(x/y))  Boole[0≤x≤(y^2) ],{x,0,1},{y,0,1}]
Plot3D[(E^(x/y)) ,{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},0<x≤(y^2) ]]

RegionPlot[{((x^2)/2)≤y≤x},{x,0,2},{y,0,2}]
Integrate[(x/((x^2)+(y^2)))  Boole[((x^2)/2)≤y≤x ],{x,0,2},{y,0,2}]
Plot3D[(x/((x^2)+(y^2))) ,{x,0,2},{y,0,2},Mesh->None,RegionFunction->Function[{x,y},((x^2)/2)≤y≤x ]]

RegionPlot[{(1/x)≤y≤x},{x,1,2},{y,(1/2),2}]
Integrate[((x^2)/(y^2))  Boole[(1/x)≤y≤x],{x,1,2},{y,(1/2),2}]
Plot3D[((x^2)/(y^2)) ,{x,1,2},{y,(1/2),2},Mesh->None,RegionFunction->Function[{x,y},(1/x)≤y≤x]]

Esercizio 18

Integrazione su domini semplici - inversione dell’ordine di integrazione

RegionPlot[{y≤x≤(E^y)},{x,0,E},{y,0,1}]
Integrate[(Sqrt[x])  Boole[y≤x≤(E^y)],{x,0,E},{y,0,1}]
Plot3D[(Sqrt[x]) ,{x,0,E},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},y≤x≤(E^y)]]

RegionPlot[{3 y≤x≤3},{x,0,3},{y,0,1}]
Integrate[(E^(x^2))  Boole[3 y≤x≤3],{x,0,3},{y,0,1}]
Plot3D[(E^(x^2)) ,{x,0,3},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},3 y≤x≤3]]

RegionPlot[{(x^2)≤y≤1},{x,0,1},{y,0,1}]
Integrate[(x^(3 )) Sin[(y^3)]  Boole[(x^2)≤y≤1],{x,0,1},{y,0,1}]
Plot3D[(x^(3 )) Sin[(y^3)] ,{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},(x^2)≤y≤1]]

RegionPlot[{(y^2)≤x≤9},{x,0,9},{y,0,3}]
Integrate[y Cos[(x^2)]  Boole[(y^2)≤x≤9],{x,0,9},{y,0,3}]
Plot3D[y Cos[(x^2)],{x,0,9},{y,0,3},Mesh->None,RegionFunction->Function[{x,y},(y^2)≤x≤9]]

Esercizio 20

Integrazione su unione di regioni semplici

a)

RegionPlot[{y≤(Sqrt[1+x]) && -(Sqrt[1+x])≤y && y≤-(1/2) x+1 && (1/2) x-1≤y },{x,-1,2},{y,-1,1}]
Integrate[x y  Boole[y≤(Sqrt[1+x]) && -(Sqrt[1+x])≤y&& y≤-(1/2) x+1 && (1/2) x-1≤y],{x,-1,2},{y,-1,1}]
Plot3D[x y,{x,-1,2},{y,-1,1},Mesh->None,RegionFunction->Function[{x,y},y≤(Sqrt[1+x]) && -(Sqrt[1+x])≤y && y≤-(1/2) x+1 && (1/2) x-1≤y ]]

RegionPlot[{0≤ y≤(Sqrt[1+x])  },{x,-1,0},{y,0,1}]
Integrate[x y  Boole[0≤ y≤(Sqrt[1+x])],{x,-1,0},{y,0,1}]
Plot3D[x y,{x,-1,0},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},0≤ y≤(Sqrt[1+x]) ]]

RegionPlot[{0≤ y≤-(1/2) x+1  },{x,0,2},{y,0,1}]
Integrate[x y  Boole[0≤ y≤-(1/2) x+1],{x,0,2},{y,0,1}]
Plot3D[x y,{x,0,2},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},0≤ y≤-(1/2) x+1 ]]

b)

RegionPlot[{y≤-x-1 ||x+1≤y || y≤x-1|| -x+1≤y },{x,-1,1},{y,-1,1}]
Integrate[x y  Boole[y≤-x-1 ||x+1≤y || y≤x-1|| -x+1≤y],{x,-1,1},{y,-1,1}]
Plot3D[x y,{x,-1,1},{y,-1,1},Mesh->None,RegionFunction->Function[{x,y},y≤-x-1 ||x+1≤y || y≤x-1|| -x+1≤y ]]

Esercizio 8.2

RegionPlot[{ 0≤ y≤1-x},{x,0,1},{y,0,1}]
Integrate[((x^2)+Sin[y]) Boole[0≤y≤1-x],{x,0,1},{y,0,1}]
Plot3D[(x^2)+Sin[y],{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y},0≤y≤1-x]]

Esercizio 8.3

RegionPlot[{ Min[(x^(1/3)),(x^3)]≤y≤Max[(x^(1/3)),(x^3)]},{x,0,2},{y,0,8}]
Integrate[(x^5) Boole[Min[(x^(1/3)),(x^3)]≤y≤Max[(x^(1/3)),(x^3)]],{x,0,2},{y,0,8}]
Plot3D[(x^5),{x,0,2},{y,0,8},Mesh->None,RegionFunction->Function[{x,y},Min[(x^(1/3)),(x^3)]≤y≤Max[(x^(1/3)),(x^3)]]]

FullSimplify[-(3/19)  (2^(19/3))+((2^9)/9)+(6/19)-(2/9)]

Esercizio 8.4

RegionPlot[{ y≤ (x/2) && y≤(Sqrt[16-(x^2)])  },{x,0,4},{y,0,(4/(Sqrt[5]))}]
Integrate[(x^2) y Boole[ y≤ (x/2) && y≤(Sqrt[16-(x^2)])],{x,0,4},{y,0,(4/(Sqrt[5]))}]
Plot3D[(x^2) y,{x,0,4},{y,0,(4/(Sqrt[5]))},Mesh->None,RegionFunction->Function[{x,y}, y≤ (x/2) && y≤(Sqrt[16-(x^2)])]]
FullSimplify[(1/40) ((((8/(Sqrt[5]))))^5)+(8/3) (4^3)-(8/3) ((((8/(Sqrt[5]))))^3)-((4^5)/10)+(1/10) ((((8/(Sqrt[5]))))^5) ]

Esercizio 8.5

RegionPlot[{(3/x)≤y≤(Sqrt[10-(x^2)]) },{x,1,3},{y,1,3}]
Integrate[((2 x  y)/(((1+(y^2)))^2)) Boole[(3/x)≤y≤(Sqrt[10-(x^2)])],{x,1,3},{y,1,3}]
Plot3D[(x^2) y,{x,1,3},{y,1,3},Mesh->None,RegionFunction->Function[{x,y}, (3/x)≤y≤(Sqrt[10-(x^2)])]]

Esercizio 8.7

RegionPlot[{(Sqrt[y])≤x≤1 },{x,0,1},{y,0,1}]
Integrate[(x^(3 )) Cos[x y] Boole[(Sqrt[y])≤x≤1],{x,0,1},{y,0,1}]
Plot3D[(x^(3 )) Cos[x y],{x,0,1},{y,0,1},Mesh->None,RegionFunction->Function[{x,y}, (Sqrt[y])≤x≤1]]

Esercizio 8.8

RegionPlot[{0≤x≤(y^2) },{x,0,Pi},{y,0,(Sqrt[Pi])}]
Integrate[((Sin[(y^2)])/y) Boole[0≤x≤(y^2) ],{x,0,Pi},{y,0,(Sqrt[Pi])}]
Plot3D[((Sin[(y^2)])/y),{x,0,Pi},{y,0,(Sqrt[Pi])},Mesh->None,RegionFunction->Function[{x,y},0≤x≤(y^2) ]]

Esercizio 8.9

GraphicsGrid[{{RegionPlot[{1-(Sqrt[1-(x^2)])≤y≤2 x },{x,0,(4/5)},{y,0,(8/5)}],RegionPlot[{1-(Sqrt[1-(x^2)])≤y≤1+(Sqrt[1-(x^2)]) },{x,(4/5),1},{y,(2/5),(8/5)}]}}]

RegionPlot[{(y/2)≤x≤(Sqrt[2 y-(y^2)]) },{x,0,1},{y,0,(8/5)}]
Integrate[((x Sin[y])/y) Boole[(y/2)≤x≤(Sqrt[2 y-(y^2)]) ],{x,0,1},{y,0,(8/5)}]
Plot3D[((x Sin[y])/y),{x,0,1},{y,0,(8/5)},Mesh->None,RegionFunction->Function[{x,y},(y/2)≤x≤(Sqrt[2 y-(y^2)]) ]]