实验二-蒙特卡罗方法计算三维体积

数学与应用数学系201 3 ～201 4 学年第二学期实验报告

(*1冰淇淋锥的体积*)

t1=ParametricPlot3D[{r*Cos[t],r*Sin[t],r^2},{t,0,2Pi},{r,0,1},DisplayFunction Identity];

t2=ParametricPlot3D[{Cos[u]*Sin[v],Sin[u]*Sin[v],1+Cos[v]},{u,0,2Pi},{v,0,Pi/ 2},DisplayFunction Identity];

Show[t1,t2,DisplayFunction$DisplayFunction]

1

0.5

-0.5

-1

2

1.5

1

0.5

-1

-0.5

0.5

1

Graphics3D

p=Complex{};

Do[m=0;

Do[x=2*Random[Real,{0,1}]-1;y=2*Random[Real,{0,1}]-1;z=2*Random[Real,{0,1}]; R1=x^2+y^2;

R2=Sqrt[R1];

If[z R2&& (z-1)^21-R1,m++],{k,1,n}];

AppendTo[p,N[8m/n]],{t,1,10}];

Print[p];

Sum[p[[t]],{t,1,10}]/10

{,,,,,,,,,}

(*1冰淇淋锥的体积*)

x=r*Sin[u]*Cos[v];

y=r*Sin[u]*Sin[v];

z=r*Cos[u]+1;

s=Integrate[r^2*Sin[u],{v,0,2Pi},{u,0,Pi/4},{r,0,2Cos[u]}];

N[s]

(*2体积*)

s1=ParametricPlot3D[{r*Sin[u],r*Cos[u],r},{u,0,2Pi},{r,0,1},DisplayFunction Identity];

t1=ParametricPlot3D[{r*Sin[u],r*Cos[u],1},{u,0,2Pi},{r,0,1},DisplayFunction Identity];

Show[s1,t1,DisplayFunction $DisplayFunction]

1-1-0.500.5

10

0.25

0.5

0.75

1

-1

-0.5

0.5

Graphics3D

(*2体积*)

n=1000;

p=Complex{};

Do[m=0;

Do[x=2*Random[Real,{0,1}]-1;y=2*Random[Real,{0,1}]-1;z=Random[Real,{0,1}];

u=2*Random[Real,{0,1}];

R1=x^2+y^2;

R2=Sqrt[R1];

If[z1&& z R2 && u R1+z^2,m++],{k,1,n}];

AppendTo[p,N[8m/n]],{t,1,10}];

Print[p];

Sum[p[[t]],{t,1,10}]/10

{,,,,,,,,,}

(*2体积*)

s=Integrate[x^2+y^2+z^2,{y,-1,1},{x,-Sqrt[1-y^2],Sqrt[1-y^2]},{z,Sqrt[x^2+y^2 ],1}];

N[s]

(*3体积*)

s=Plot[x-2,{x,1,4},DisplayFunction Identity];

t=Plot[Sqrt[x],{x,1,4},DisplayFunction Identity];

Show[s,t,DisplayFunction$DisplayFunction]

2

1.5

1

0.5

1.52

2.53

3.54

-0.5

-1

Graphics

n=10000;(*3体积*)

p=Complex{};

Do[m=0;

Do[x=4*Random[Real,{0,1}];y=3*Random[Real,{0,1}]-1;z=16*Random[Real,{0,1}]; If[x y^2&& x y+2 &&z x*y^2,m++],{k,1,n}];

AppendTo[p,N[192*m/n]],{t,1,10}];

Print[p];

Sum[p[[t]],{t,1,10}]/10

{,,,,,,,,,}

(*3体积*)

s=Integrate[x*y^2,{x,1,4},{y,x-2,Sqrt[x]}];

N[s]