Mehrdimensionale Integration

RechtsRiemannSumme[f_,{x_,y_},{{a_,b_},{c_,d_}}]:=(((b-a)(d-c))/(n^2))(∑_(k=1)^n)(∑_(j=1)^n)(f/.{x->a+(k/n)(b-a),y->c+(j/n)(d-c)})

riemann=RechtsRiemannSumme[(x+y)^3,{x,y},{{0,1},{0,1}}]

Limit[riemann,n->∞]

riemann=RechtsRiemannSumme[(x+y)^3,{x,y},{{a,b},{c,d}}]

Limit[riemann,n->∞]//Factor

Vergleich

(∫_0^1)(∫_0^1)(((x+y))^3)\[DifferentialD]x\[DifferentialD]y

(∫_c^d)(∫_a^b)(((x+y))^3)\[DifferentialD]x\[DifferentialD]y

(∫_a^b)(∫_c^d)(((x+y))^3)\[DifferentialD]x\[DifferentialD]y//Factor

Beispiel 9.2

(∫_(-1)^1)(∫_(-2)^2)((Subscript[x, 1])^2)((Subscript[x, 2])^4)\[DifferentialD](Subscript[x, 2])\[DifferentialD](Subscript[x, 1])

Beispiel 9.3

(∫_0^2)((∫_(-1)^1)(∫_0^1)((Subscript[x, 1])^2)((Subscript[x, 2])^2)((Subscript[x, 3])^2)\[DifferentialD](Subscript[x, 1])\[DifferentialD](Subscript[x, 2]))\[DifferentialD](Subscript[x, 3])

Beispiel 9.4

(∫_(-1)^1)(∫_0^(Sqrt[1-((Subscript[x, 1])^2)]))1\[DifferentialD](Subscript[x, 2])\[DifferentialD](Subscript[x, 1])

∫((∫_0^(Sqrt[1-((Subscript[x, 1])^2)]))1\[DifferentialD](Subscript[x, 2]))\[DifferentialD](Subscript[x, 1])

Beispiel 9.5

π(∫_0^H)((R^2)/(H^2))(((H-(Subscript[x, 3])))^2)\[DifferentialD](Subscript[x, 3])

Beispiel 9.6

V=Factor[(∫_0^(b-(b/c)(Subscript[x, 3])))(∫_0^(a-(a/b)(Subscript[x, 2])-(a/c)(Subscript[x, 3])))1\[DifferentialD](Subscript[x, 1])\[DifferentialD](Subscript[x, 2])]

(∫_0^c)V\[DifferentialD](Subscript[x, 3])

Beispiel 9.7

(∫_0^a)(∫_0^(Sqrt[(b^2)-((b^2)/(a^2))((Subscript[x, 1])^2)]))(Subscript[x, 2])\[DifferentialD](Subscript[x, 2])\[DifferentialD](Subscript[x, 1])

(∫_0^b)(∫_0^(Sqrt[(a^2)-((a^2)/(b^2))((Subscript[x, 2])^2)]))(Subscript[x, 2])\[DifferentialD](Subscript[x, 1])\[DifferentialD](Subscript[x, 2])

Beispiel 9.8

(∫_0^1)(∫_(x^2)^(-(((x-1))^2)+1))x\[DifferentialD]y\[DifferentialD]x

(∫_0^1)(∫_(-(Sqrt[1-y])+1)^(Sqrt[y]))x\[DifferentialD]x\[DifferentialD]y

Beispiel 9.10

8(∫_0^1)((∫_0^(Sqrt[1-(x^2)]))(∫_0^(Sqrt[1-(x^2)]))1\[DifferentialD]z\[DifferentialD]y)\[DifferentialD]x

plot1=ContourPlot3D[(x^2)+(y^2)==1,{x,-2,2},{y,-2,2},{z,-2,2}]

plot2=ContourPlot3D[(x^2)+(z^2)==1,{x,-2,2},{y,-2,2},{z,-2,2}]

Show[plot1,plot2]

Plot[Exp[-(x^2)],{x,-5,5}]

∫Exp[-(x^2)]\[DifferentialD]x

(∫_(-∞)^∞)Exp[-(x^2)]\[DifferentialD]x

Plot3D[Exp[-(x^2)-(y^2)],{x,-5,5},{y,-5,5},PlotRange->All]

Jacobimatrix

JacobiMatrix[f_,var_]:=Table[D[f[[k]],var[[j]]],{k,Length[f]},{j,Length[var]}]

Zylinderkoordinaten

zylinder=ParametricPlot3D[{Cos[ϕ],Sin[ϕ],z},{ϕ,0,2π},{z,-1,1}]

jacobi=JacobiMatrix[{r Cos[ϕ],r Sin[ϕ],z},{r,ϕ,z}]

Det[jacobi]//Simplify

Beispiel 9.11

(∫_0^(2π))(∫_0^r)r\[DifferentialD]r\[DifferentialD]ϕ

Beispiel 9.12

int1=(∫_0^(2π))(∫_0^R)r Exp[-(r^2)]\[DifferentialD]r\[DifferentialD]ϕ

Limit[int1,R->∞]

int2=(∫_(-S)^S)(∫_(-S)^S)Exp[-(x^2)-(y^2)]\[DifferentialD]x\[DifferentialD]y

Limit[int2,S->∞]

Plot3D[Exp[-(x^2)-(y^2)],{x,-3,3},{y,-3,3},PlotRange->All]

Beispiel 9.13

int=(∫_0^(2π))(∫_0^H)(∫_0^((R/H)z))(r^3)\[DifferentialD]r\[DifferentialD]z\[DifferentialD]ϕ

Beispiel 9.14

Plot3D[(Sqrt[(x^2)+(y^2)]),{x,-3,3},{y,-3,3}]

Plot3D[1+(Sqrt[1-((x^2)+(y^2))]),{x,-3,3},{y,-3,3}]

int=(∫_(-1)^1)(∫_(-(Sqrt[1-(x^2)]))^(Sqrt[1-(x^2)]))(∫_(Sqrt[(x^2)+(y^2)])^(1+(Sqrt[1-((x^2)+(y^2))])))\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x

inner1=(∫_(Sqrt[(x^2)+(y^2)])^(1+(Sqrt[1-((x^2)+(y^2))])))\[DifferentialD]z

inner2=Integrate[1,{y,-(Sqrt[1-(x^2)]),(Sqrt[1-(x^2)])},{z,(Sqrt[(x^2)+(y^2)]),1+(Sqrt[1-((x^2)+(y^2))])},GenerateConditions->False]

int=(∫_0^(2π))(∫_0^1)(∫_r^(1+(Sqrt[1-(r^2)])))r \[DifferentialD]z\[DifferentialD]r\[DifferentialD]ϕ

inner1=(∫_r^(1+(Sqrt[1-(r^2)])))r \[DifferentialD]z

inner2=(∫_0^1)(∫_r^(1+(Sqrt[1-(r^2)])))r \[DifferentialD]z\[DifferentialD]r

Kugelkoordinaten

halbkugel=ParametricPlot3D[{Cos[ϕ]Sin[θ],Sin[ϕ]Sin[θ],Cos[θ]},{ϕ,0,2π},{θ,0,(π/2)}]

halbkugel=ParametricPlot3D[{Cos[ϕ]Cos[θ],Sin[ϕ]Cos[θ],Sin[θ]},{ϕ,0,2π},{θ,0,(π/2)}]

kugel=ParametricPlot3D[{Cos[ϕ]Cos[θ],Sin[ϕ]Cos[θ],Sin[θ]},{ϕ,0,2π},{θ,-(π/2),(π/2)}]

jacobi=JacobiMatrix[{r Cos[ϕ]Cos[θ],r Sin[ϕ]Cos[θ],r Sin[θ]},{r,ϕ,θ}]

Det[jacobi]//Simplify

Beispiel 9.15

int=(∫_0^π)(∫_0^(2π))(∫_0^R)(r^2)Sin[θ]\[DifferentialD]r\[DifferentialD]ϕ\[DifferentialD]θ

Beispiel 9.16

int=(∫_R0^R)(∫_0^(2π))(∫_0^(ArcCos[(R0/r)]))(r^4)Sin[θ]\[DifferentialD]θ\[DifferentialD]ϕ\[DifferentialD]r

Beispiel 9.17

int=c(∫_0^1)(∫_0^(π/2))(Sqrt[1-(r^2)])a b r\[DifferentialD]ϕ\[DifferentialD]r

Übung 9.10

plot1=ContourPlot3D[(x^2)+(y^2)+(z^2)==8,{x,-3,3},{y,-3,3},{z,0,3},AspectRatio->(1/2)]

plot2=Plot3D[(((x^2)+(y^2)+4)/4),{x,-3,3},{y,-3,3}]

Show[plot1,plot2]

int=(∫_0^(2π))(∫_0^2)(∫_(((r^2)/4)+1)^(Sqrt[8-(r^2)]))r\[DifferentialD]z\[DifferentialD]r\[DifferentialD]ϕ

Mittlerer Abstand vom Ursprung im Einheitswürfel

int=(∫_0^1)(∫_0^1)(∫_0^1)(Sqrt[(x^2)+(y^2)+(z^2)])\[DifferentialD]x\[DifferentialD]y\[DifferentialD]z

FullSimplify[int]

N[int]