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=0,1×0,1.解In1:= Integratex3+3x2y+y3,x,0,1,y,0,1Out1= 12. (p,q是常数), 其中R=0,a&#

2、215;0,a.解In1:= 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×-3,3×0,1.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,

3、0,3,DisplayFunction->Identity; 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,

4、x,ySolvex*y-1= =0,y-x= =0,x,yOut2= x->2,y->2 x->,y->2 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,

5、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=Plot3D0,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,DisplayFuncti

6、on->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; Showb1,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

7、, DisplayFunction->Identity; t2=Plot3D4-x-y, x,0,4,y,0,4, DisplayFunction->Identity;t3=ParametricPlot3D2,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,DisplayFu

8、nction->$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,DisPlayFunction->Identity; a2=PrametricPlot2,y,y,0,4,DisPlayFunction->Identity;a3=Plot4-x,x,0,4,DisPlayFunction->Identity;Showa1,a2,a3,AxesLabel->“x”,”y

9、”,AspectRatio->Automatic,PlotRange->0,4,DisplayFunction->$DisplayFunction Out2= -Graphics-In3:= b1=Integrate4-x-y,x,0,1,y,0,3;b2=Integrate4-x-y,x,1,2,y,0,4;b3=b1+b2 Out3= In4:= c1=Plot3Dx*y,x,-2,2,y,-2,2,DisplayFunction->Identity;c2=ParametricPlot3D(1/2)Sint+1/2,(1/2)Cost,z,t,0,2Pi,z,-2,2,DisplayFunction->$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) ,其中=3