用Mathematica求重积分以及相关的应用

1、5 用Mathematica求重积分以及相关的应用5.1 常用的重积分运算函数ParametricPlot xt,yt,t,a,b: 作二维参数方程的图形。Plot3Dfx,y,x,a,b,y,c,d: 作的图形。ParametricPlot3Dxu,v,yu,v,zu,v,u,a,bv,c,d: 作三维参数方程的图形。Integratefx,y,x,a,b,y,c,d: 计算累次积分。例5.1 计算下列重积分：1. , 其中R=0,10,1. 解 In1:= Integratex3+3x2y+y3,x,0,1,y,0,1Out1= 12. (p,q是常数), 其中R=0,a0,a. 解 In

2、1:= IntegrateE(p*x+q*y),x,0,a,y,0,aOut1= 3. , 其中R=-1,1-1,1. 解 In1:= IntegrateAbsx+y,x,0,Pi,y,0,PiOut1= 4. ,其中V=-2,5-3,30,1. 解 In1:= Integratex*y+z2,x,-2,5,y,-3,3,z,0,1Out1= 14例5.2 计算下列重积分：1. 求二重积分，其中是D由直线x=2,y=x和xy=1双曲线所围成。 解 先画出被积区域D的图形:In1:= Cleart1,t2; a=ParametricPlot2,y,y,0,3,DisplayFunxtion-Id

3、entity; b=Ploty=x,y=1/x,PlotRange-0,3,AspectRatio-Automatic,DisplayFunction-Indentity; Showa,b, PlotRange-0,2.5,AspectRatio-Automatic,DisplayFunction-$DisplayFunction;Out1= -Graphics-再求出D的边界曲线的交点:In2:= Solvex-2= =0,y-x= =0,x,y Solvex-2= =0,x*y-1= =0,x,ySolvex*y-1= =0,y-x= =0,x,yOut2= x-2,y-2 x-,y-2

4、x-1,y-1,x-1,y-1最后计算积分:In3:= Cleary;Integratex2/y2,x,1,2,y,1/x,xOut3= 2. 求二重积分，其中D是. 解 先画出被积区域的图形:In1:=ParametricPlot(1/2)*Sint+1/2,(1/2)*Cost,t,0,2Pi,AspectRatio-AutomaticOut1= -Graphics-计算积分:In2:= IntegrateSqrtx,x,0,1,y,0,Sqrtx-x2Out2= 3. 求三重积分，其中V是由曲面z=xy,平面y=x， x=1，z=0所围成。 解 先画出空间图形:In1:= a1=Plot

5、3D0,x,0,2,y,0,2,DisplayFunction-Identity;a2=Plot3Dx*y,x,0,2,y,0,2,DisplayFunction-Identity; a3=ParametricPlot3Dx,x,zx,0,2,z,0,2,DisplayFunction-Identity; a4=ParametricPlot3D1,y,zy,0,2,z,0,2,DisplayFunction-Identity;Out1= -Graphics-再画出立体关于XOY面的投影域:In2:= b1=Plotx,-2,2;b2=ParametricPlot1,y,y,-2,2; Show

6、b1,b2Out2= -Graphics-计算积分:In3:= Clearx,y,z; Integratex*y2*z3,x,0,1,y,0,x,z,0,x*yOut3= 例5.3 求由坐标平面及x=2, y=3, x+y+z=4，所围的角柱体的体积。 解 In1:= Clearx,y,u,v,t1,t2,t3,t4,a1,a2,a3,a4,b1,b2; t1=Plot3D0,x,0,4,y,0,4, DisplayFunction-Identity; t2=Plot3D4-x-y, x,0,4,y,0,4, DisplayFunction-Identity;t3=ParametricPlot

7、3D2,y,z,y,0,4,z,0,4,DisplayFunction-Identity;t4=ParametricPlot3Dx,3,z,x,0,4,y,0,4, DisplayFunction-Identity;Showt1,t2,t3,t4,AxesLabel-“x”,”y”,”z”,AspectRatio-Automatic,DisplayFunction-$DisplayFunction,PlotRange-0,4,ViewPiont-1,-1,1Out1= -Graphics-例5.4 求z=xy,围成的体的体积。 解 In2:= a1=PrametricPlotx,3,x,0,4

8、,DisPlayFunction-Identity; a2=PrametricPlot2,y,y,0,4,DisPlayFunction-Identity;a3=Plot4-x,x,0,4,DisPlayFunction-Identity;Showa1,a2,a3,AxesLabel-“x”,”y”,AspectRatio-Automatic,PlotRange-0,4,DisplayFunction-$DisplayFunction Out2= -Graphics-In3:= b1=Integrate4-x-y,x,0,1,y,0,3;b2=Integrate4-x-y,x,1,2,y,0,

9、4;b3=b1+b2 Out3= In4:= c1=Plot3Dx*y,x,-2,2,y,-2,2,DisplayFunction-Identity; c2=ParametricPlot3D(1/2)Sint+1/2,(1/2)Cost,z,t,0,2Pi,z,-2,2,DisplayFunction-$DisplayFunction,ViewPoint-1,2,2 Out4= -Graphics-In5:=ParametricPlot(1/2)*Sint+1/2,(1/2)*Cost,t,0,2Pi,AspectRatio-Automatic Out5= -Graphics- In6:= Integratex*y,x,0,1,y,0,Sqrtx-x2 Out6:= In7:= %*2 Out7= 练习9.51 计算下列重积分：(1)