用Mathematica求重积分以及相关的应用

1、-PAGE . z.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 *t,yt,t,a,b:作二维参数方程的图形。Plot3Df*,y,*,a,b,y,c,d: 作的图形。ParametricPlot3D*u,v,yu,v,zu,v,u,a,bv,c,d: 作三维参数方程的图形。Integratef*,y,*,a,b,y,c,d: 计算累次积分。例5.1计算以下重积分：1., 其中R=0,10,1.解In1:= Integrate*3+3*2y+y3,*,0,1,y,0,1Out1= 12. (p,q是常数), 其中R=0,a0,a.

2、解In1:= IntegrateE(p*+q*y),*,0,a,y,0,aOut1= 3. , 其中R=-1,1-1,1.解In1:= IntegrateAbs*+y,*,0,Pi,y,0,PiOut1= 4. ,其中V=-2,5-3,30,1.解In1:= Integrate*y+z2,*,-2,5,y,-3,3,z,0,1Out1= 14例5.2计算以下重积分：求二重积分，其中是D由直线*=2,y=*和*y=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,DisplayFun*tion-Identity; b

3、=Ploty=*,y=1/*,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:= Solve*-2= =0,y-*= =0,*,y Solve*-2= =0,*y-1= =0,*,ySolve*y-1= =0,y-*= =0,*,yOut2= *-2,y-2 *-,y-2 *-1,y-1,*-1

4、,y-1最后计算积分:In3:= Cleary;Integrate*2/y2,*,1,2,y,1/*,*Out3= 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:= IntegrateSqrt*,*,0,1,y,0,Sqrt*-*2Out2= 求三重积分，其中V是由曲面z=*y,平面y=*， *=1，z=0所围成。解先画出空间图形:In1:= a1=Plot3D0,*,0,2,y,0,2,Disp

5、layFunction-Identity;a2=Plot3D*y,*,0,2,y,0,2,DisplayFunction-Identity; a3=ParametricPlot3D*,*,z*,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*OY面的投影域:In2:= b1=Plot*,-2,2;b2=ParametricPlot1,y,y,-2,2; Showb1,b2Out2= -Graphics-

6、计算积分:In3:= Clear*,y,z; Integrate*y2*z3,*,0,1,y,0,*,z,0,*yOut3= 例5.3 求由坐标平面及*=2, y=3, *+y+z=4，所围的角柱体的体积。解In1:= Clear*,y,u,v,t1,t2,t3,t4,a1,a2,a3,a4,b1,b2; t1=Plot3D0,*,0,4,y,0,4, DisplayFunction-Identity; t2=Plot3D4-*-y, *,0,4,y,0,4, DisplayFunction-Identity;t3=ParametricPlot3D2,y,z,y,0,4,z,0,4,Displ

7、ayFunction-Identity;t4=ParametricPlot3D*,3,z,*,0,4,y,0,4, DisplayFunction-Identity;Showt1,t2,t3,t4,A*esLabel-*,y,z,AspectRatio-Automatic,DisplayFunction-$DisplayFunction,PlotRange-0,4,ViewPiont-1,-1,1Out1= -Graphics-例5.4 求z=*y,围成的体的体积。解 In2:= a1=PrametricPlot*,3,*,0,4,DisPlayFunction-Identity; a2=Pr

8、ametricPlot2,y,y,0,4,DisPlayFunction-Identity;a3=Plot4-*,*,0,4,DisPlayFunction-Identity;Showa1,a2,a3,A*esLabel-*,y,AspectRatio-Automatic,PlotRange-0,4,DisplayFunction-$DisplayFunction Out2= -Graphics-In3:= b1=Integrate4-*-y,*,0,1,y,0,3;b2=Integrate4-*-y,*,1,2,y,0,4;b3=b1+b2 Out3= In4:= c1=Plot3D*y,*

9、,-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:= Integrate*y,*,0,1,y,0,Sqrt*-*2 Out6:= In7:= %*2 Out7= 练习9.51 计算以下重