§8 用Mathematica求重积分以及相关的应用练习解答

1 计算下列重积分： (1)

(y2x)dxdy,其中D=[3,5]×[1,2];

D解 In[1]:= Integrate[y-2x,{x,3,5},{y,1,2}] Out[1]= -13 (2)

Cos(xy)dxdy，其中D=[0,

D]×[0,]; 2解 In[1]:= Integrate[Cos[x+y],{x,0,Pi/2},{y,0,Pi}] Out[1]= -2

(3) 域；

解 In[1]:= a=Plot[Sqrt[2x],{x,0,2},DisplayFunction->Identity];

b=ParametricPlot[{2,y},{y,0,4}, DisplayFunction->Identity]; Show[a,b,PlotRange->{0,4}, DisplayFunction->$ DisplayFunction]

43.532.521.510.50.511.5222y2px与直线x，其中由抛物线xydxdyDDp(p0)所围成的区2 Out[1]= -Graphics-

In[2]:= Integrate[x*y^2,{x,0, 2},{y,0,2px}]

16px3Out[2]=

3(4)

222xya，其中为圆域； |xy|dxdyDD 1073

解 In[1]:= ParametricPlot[{Sin[t],Cos[t],{t,0,2Pi},AspectRatio->Automatic}

10.5-1-0.50.51-0.5-1 Out[1]= -Graphics-

In[2]:= Integrate[Abs[x*y],{x,-1,1},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2] }] Out[2]= (5)

 2]×[0,]; 22xcosycoszdxdydz，其中V=[0,1]×[0,

V解 In[1]:= Integrate[x*Cos[y]*Cos[z],{x,0,1},{y,0,Pi/2 },{ z,0,Pi/2 }] Out[1]= (6)

1 2ycos(xz)dxdydz，其中V是由yx，y=0，z=0及x+z=V所2围成的区域。

解 In[1]:= a=ParametricPlot3D[{x,y,Pi/2-x},{x,0,2},{y,0,2},

DisplayFunction->Identity];

b=ParametricPlot3D[{x,Sqrt[x],z},{x,0,2},{y,0,2},

DisplayFunction->Identity];

c=Plot3D[0,{x,0,2},{y,0,2},DisplayFunction->Identity]; d= Plot3D [0,{x,0,2},{z,0,2}, DisplayFunction->Identity];

Show[a,b,c,d,AxesLabel->{“x”,”y”,”z”}, AspectRatio->Automatic,

PlotRange->{0,2},DisplayFunction->$DisplayFunction, ViewPoint->{1,-2,0}]

1074

2 0 0.5 x 1 1.5 2 1.5 z 1 0.5 2 1.5 1 0.5y 0

Out[1]= -Graphics3D-

0 In[2]:= Integrate[y*Cos[x+z],{x,0, Pi/2},{y,0, Sqrt[x]},{ z,0,Pi-x}] Out[2]= 1 22 求下列曲面所围成立体V的体积。

(1) V是由zx2y2和z=x+y所围成的立体； 解 In[1]:= a=Plot3D[x^2+y^2,{x,-2,2},{y,-2,2},

DisplayFunction->Identity];

b=Plot3D[x+y,{x,-1,2},{y,-1,2},

DisplayFunction->Identity];

Show[a,b,AxesLabel->{“x”,”y”,”z”},PlotRange->{-2,5},

DisplayFunction->$DisplayFunction,ViewPoint->{0,2,0} Shading->False]

210-1-2y420210x-1-2-2 zOut[1]= -Graphics3D-

1075

In[2]:= x=r*Cos[t]+1/4

y=r*Sin[t]+1/4

Integrate[x^2+y^2-x-y,{x,0, Pi/2},{y,0,1/2}]

Out[2]=

 8(2) V是由曲面z22x2y2和zx2y2所围成的立体； 解 In[1]:= a=Plot3D[2-x^2-y^2,{x,-1,1},{y,-1,1},

DisplayFunction->Identity];

b= ParametricPlot3D[{u*Cos[v],u*Sin[v],u^2},{u,-1,2},{v,-Pi,Pi},

DisplayFunction->Identity];

Show[a,b,AxesLabel->{“x”,”y”,”z”},PlotRange->{0,2},

DisplayFunction->$DisplayFunction]

21.5z10.500-20x2-22y Out[1]= -Graphics3D-

In[2]:= ParametricPlot[{Sin[t],Cos[t]},{t,0,2Pi}, AxesLabel->{“x”,”y”,”z”},

AspectRatio->Automatic]

1076

y10.5-1-0.50.51x-0.5-1 Out[2]= -Graphics-

In[3]:= Integrate[2-x^2-y^2,{x,0,1},{y,0,Sqrt[1-x^2]}] Out[3]= 4

1077