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1、牟合方盖 1,用切片法讨论牟合方盖,蜀南竹海,牟合方盖 2,牟合方盖就是两个半径相同的直交圆柱面所围成的立体。 教材中一般是利用二重积分计算其体积。 本课件用截面的面积的定积分来计算其体积。 还用数学软件Maple制作了有关动画。 最后比较牟合方盖与另一个立体的体积。,牟合方盖 3,求两直交圆柱面,所围成的立体的体积,牟合方盖 4,牟合方盖,牟合方盖 5,with(plots):R:=1: x_axis:=plot3d(u,0,0,u=01.5,v=00.01,thickness=3): y_axis:=plot3d(0,u,0,u=01.3,v=00.01,thickness=3): z_a
2、xis:=plot3d(0,0,u,u=01.3,v=00.01,thickness=3): zuobiaoxi:=display(x_axis,y_axis,z_axis): zhumian1:=plot3d(R*cos(t),R*sin(t),z,z=0R*sin(t),t=0Pi/2,color=yellow): quxian1:=spacecurve(R*cos(t),R*sin(t),R*sin(t),t=0Pi/2,color=red,thickness=5): quxian2:=spacecurve(R*cos(t),R*sin(t),0,t=0Pi/2,color=blue,t
3、hickness=5): zhumian2:=plot3d(R*cos(t),y,R*sin(t),y=0R*sin(t),t=0Pi/2,color=green): display(zhumian1,zhumian2,zuobiaoxi,quxian1,quxian2,scaling=constrained,orientation=28,53);,牟合方盖 6,牟合方盖,刘徽在他的九章算术注中，提出一个独特的方法来计算球体的体积：他不直接求球体的体积，而是先计算另一个叫牟合方盖的立体的体积。 所谓牟合方盖，就是指由两个同样大小但轴心互相垂直的圆柱体相交而成的立体。由于这立体的外形似两把上下对
4、称的正方形雨伞，所以就称它为牟合方盖。 在这个立体里面，可以内切一个半径和原本圆柱体一样大小的球体，刘徽并指出，由于内切圆的面积和外切正方形的面积之比为:4，所以牟合方盖的体积与球体体积之比亦应为:4。 可惜的是，刘徽并没有求出牟合方盖的体积，所以亦不知道球体体积的计算公式。,http:/ 7,下面用截面来研究牟合方盖,牟合方盖 8,牟合方盖 9,从 x 轴正向看去,牟合方盖 10,with(plots): R:=1: f:=x-sqrt(R2-x2): a:=-R:b:=R: xzou:=spacecurve(x,0,0,x=a-1b+1,thickness=3,color=black):
5、yzou:=spacecurve(0,y,0,y=a-1b+1,thickness=3,color=black): zzou:=spacecurve(0,0,z,z=a-1b+1,thickness=3,color=black): K:=60:for i from 0 to K do xi:=a+i*(b-a)/K: zhengfangxingi:=spacecurve(xi,f(xi),f(xi),xi,-f(xi),f(xi),xi,-f(xi),-f(xi),xi,f(xi),-f(xi),xi,f(xi),f(xi),thickness=3,color=blue): zhengfang
6、bani:=plot3d(xi,y,z,y=-f(xi)f(xi),z=-f(xi)f(xi),color=grey,style=patchnogrid): qumian1i:=plot3d(x,y,f(x),x=axi,y=-f(x)f(x),color=green): qumian2i:=plot3d(x,y,-f(x),x=axi,y=-f(x)f(x),color=green): qumian3i:=plot3d(x,f(x),z,x=axi,z=-f(x)f(x),color=yellow): qumian4i:=plot3d(x,-f(x),z,x=axi,z=-f(x)f(x),
7、color=yellow)od: zhengfangxing:=display(seq(zhengfangxingi,i=0K),insequence=true): zhengfangban:=display(seq(zhengfangbani,i=0K),insequence=true): qumian1:=display(seq(qumian1i,i=0K),insequence=true): qumian2:=display(seq(qumian2i,i=0K),insequence=true): qumian3:=display(seq(qumian3i,i=0K),insequenc
8、e=true): qumian4:=display(seq(qumian4i,i=0K),insequence=true): display(xzou,yzou,zzou,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,qumian4,scaling=constrained);,动画的Maple程序,牟合方盖 11,with(plots): R:=1: f:=x-sqrt(R2-x2): a:=-R:b:=R: xzou:=spacecurve(x,0,0,x=a-1/2b+1/2,thickness=3,color=black): yzo
9、u:=spacecurve(0,y,0,y=a-1/2b+1/2,thickness=3,color=black): zzou:=spacecurve(0,0,z,z=a-1/2b+1/2,thickness=3,color=black): yuan1:=spacecurve(R*cos(t),R*sin(t),0,t=02*Pi,thickness=3,color=red): yuan2:=spacecurve(R*cos(t),0,R*sin(t),t=02*Pi,thickness=3,color=red): xi:=1*R: zhengfangxing:=spacecurve(xi,f
10、(xi),f(xi),xi,-f(xi),f(xi),xi,-f(xi),-f(xi),xi,f(xi),-f(xi),xi,f(xi),f(xi),thickness=3,color=blue): zhengfangban:=plot3d(xi,y,z,y=-f(xi)f(xi),z=-f(xi)f(xi),color=grey,style=patchnogrid): qumian1:=plot3d(x,y,f(x),x=axi,y=-f(x)f(x),color=green): qumian2:=plot3d(x,y,-f(x),x=axi,y=-f(x)f(x),color=green)
11、: qumian3:=plot3d(x,f(x),z,x=axi,z=-f(x)f(x),color=yellow): qumian4:=plot3d(x,-f(x),z,x=axi,z=-f(x)f(x),color=yellow): display(xzou,yzou,zzou,yuan1,yuan2,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,qumian4,scaling=constrained,orientation=60,73);,牟合方盖 12,牟合方盖 13,with(plots): R:=1: f:=x-sqrt(R2
12、-x2): a:=-R:b:=R: xzou:=spacecurve(x,0,0,x=a-1b+1,thickness=3,color=black): yzou:=spacecurve(0,y,0,y=a-1b+1,thickness=3,color=black): zzou:=spacecurve(0,0,z,z=a-1b+1,thickness=3,color=black): zhumian1:=plot3d(R*cos(t),R*sin(t),z,t=02*Pi,z=a-1b+1,style=wireframe,color=blue): zhumian2:=plot3d(R*cos(t)
13、,y,R*sin(t),t=02*Pi,y=a-1b+1,style=wireframe,color=brown): K:=60:for i from 0 to K do xi:=a+i*(b-a)/K: zhengfangxingi:=spacecurve(xi,f(xi),f(xi),xi,-f(xi),f(xi),xi,-f(xi),-f(xi),xi,f(xi),-f(xi),xi,f(xi),f(xi),thickness=3,color=blue): zhengfangbani:=plot3d(xi,y,z,y=-f(xi)f(xi),z=-f(xi)f(xi),color=gre
14、y,style=patchnogrid): qumian1i:=plot3d(x,y,f(x),x=axi,y=-f(x)f(x),color=green,style=patchnogrid): qumian2i:=plot3d(x,y,-f(x),x=axi,y=-f(x)f(x),color=green,style=patchnogrid): qumian3i:=plot3d(x,f(x),z,x=axi,z=-f(x)f(x),color=yellow,style=patchnogrid): qumian4i:=plot3d(x,-f(x),z,x=axi,z=-f(x)f(x),col
15、or=yellow,style=patchnogrid)od: zhengfangxing:=display(seq(zhengfangxingi,i=0K),insequence=true): zhengfangban:=display(seq(zhengfangbani,i=0K),insequence=true): qumian1:=display(seq(qumian1i,i=0K),insequence=true): qumian2:=display(seq(qumian2i,i=0K),insequence=true): qumian3:=display(seq(qumian3i,
16、i=0K),insequence=true): qumian4:=display(seq(qumian4i,i=0K),insequence=true): display(xzou,yzou,zzou,zhumian1,zhumian2,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,qumian4,scaling=constrained);,动画的Maple程序,牟合方盖 14,下面来求牟合方盖的体积,牟合方盖 15,这与用二重积分计算的结果相同 见同济高等数学六版，下册 143页，例4,垂直于x轴的截面是一个正方形：,牟合方盖 16,牟
17、合方盖的体积与下面这个立体的体积相等,牟合方盖 17,牟合方盖 18,with(plots): R:=1.6: f:=x-sqrt(R2-x2): g:=x-sqrt(R2-x2): a:=-R:b:=R: xzou:=spacecurve(x,0,0,x=a-1b+1,thickness=3,color=black): yzou:=spacecurve(0,y,0,y=a-1b+1,thickness=3,color=black): base:=plot3d(x,y,0,x=ab,y=g(x)f(x),color=grey,style=patchnogrid): quxian:=spacec
18、urve(R*cos(t),R*sin(t),0,t=02*Pi,thickness=3,color=red): K:=60:for i from 0 to K do xi:=a+i*(b-a)/K: zhengfangxingi:=spacecurve(xi,g(xi),0,xi,f(xi),0,xi,f(xi),f(xi)-g(xi),xi,g(xi),f(xi)-g(xi),xi,g(xi),0,thickness=3,color=blue): zhengfangbani:=plot3d(xi,y,z,y=g(xi)f(xi),z=0f(xi)-g(xi),color=yellow,st
19、yle=patchnogrid): qumian1i:=plot3d(x,f(x),(f(x)-g(x)*t,t=01,x=axi,color=green): qumian2i:=plot3d(x,g(x),(f(x)-g(x)*t,t=01,x=axi,color=green): qumian3i:=plot3d(x,g(x)+(f(x)-g(x)*t,f(x)-g(x),t=01,x=axi,color=grey)od: zhengfangxing:=display(seq(zhengfangxingi,i=0K),insequence=true): zhengfangban:=displ
20、ay(seq(zhengfangbani,i=0K),insequence=true): qumian1:=display(seq(qumian1i,i=0K),insequence=true): qumian2:=display(seq(qumian2i,i=0K),insequence=true): qumian3:=display(seq(qumian3i,i=0K),insequence=true): display(xzou,yzou,base,quxian,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,scaling=constrained,orientation=-60,70);,动画的Maple程序,牟合方盖 19,这个体积刚好等于牟合方盖的体积,现在来求这个 立体的体积,牟合方盖 20,