用切片法讨论牟合方盖

牟合方盖,*,July 2, 2012,四川大学数学学院,徐小湛,用切片法讨论牟合方盖,蜀南竹海,1,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。,2,求两直交圆柱面,所围成的立体的体积,3,牟合方盖,4,with(plots):R:=1:,x_axis:=plot3d(u,0,0,u=0.1.5,v=0.0.01,thickness=3):,y_axis:=plot3d(0,u,0,u=0.1.3,v=0.0.01,thickness=3):,z_axis:=plot3d(0,0,u,u=0.1.3,v=0.0.01,thickness=3):,zuobiaoxi:=display(x_axis,y_axis,z_axis):,zhumian1:=plot3d(R*cos(t),R*sin(t),z,z=0.R*sin(t),t=0.Pi/2,color=yellow):,quxian1:=spacecurve(R*cos(t),R*sin(t),R*sin(t),t=0.Pi/2,color=red,thickness=5):,quxian2:=spacecurve(R*cos(t),R*sin(t),0,t=0.Pi/2,color=blue,thickness=5):,zhumian2:=plot3d(R*cos(t),y,R*sin(t),y=0.R*sin(t),t=0.Pi/2,color=green):,display(zhumian1,zhumian2,zuobiaoxi,quxian1,quxian2,scaling=constrained,orientation=28,53);,5,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。,6,下面用截面来研究牟合方盖,7,8,从,x,轴正向看去,9,with(plots):,R:=1:,f:=x-sqrt(R2-x2):,a:=-R:b:=R:,xzou:=spacecurve(x,0,0,x=a-1.b+1,thickness=3,color=black):,yzou:=spacecurve(0,y,0,y=a-1.b+1,thickness=3,color=black):,zzou:=spacecurve(0,0,z,z=a-1.b+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):,zhengfangbani:=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=a.xi,y=-f(x).f(x),color=green):,qumian2i:=plot3d(x,y,-f(x),x=a.xi,y=-f(x).f(x),color=green):,qumian3i:=plot3d(x,f(x),z,x=a.xi,z=-f(x).f(x),color=yellow):,qumian4i:=plot3d(x,-f(x),z,x=a.xi,z=-f(x).f(x),color=yellow)od:,zhengfangxing:=display(seq(zhengfangxingi,i=0.K),insequence=true):,zhengfangban:=display(seq(zhengfangbani,i=0.K),insequence=true):,qumian1:=display(seq(qumian1i,i=0.K),insequence=true):,qumian2:=display(seq(qumian2i,i=0.K),insequence=true):,qumian3:=display(seq(qumian3i,i=0.K),insequence=true):,qumian4:=display(seq(qumian4i,i=0.K),insequence=true):,display(xzou,yzou,zzou,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,qumian4,scaling=constrained);,动画的,Maple,程序,10,with(plots):,R:=1:,f:=x-sqrt(R2-x2):,a:=-R:b:=R:,xzou:=spacecurve(x,0,0,x=a-1/2.b+1/2,thickness=3,color=black):,yzou:=spacecurve(0,y,0,y=a-1/2.b+1/2,thickness=3,color=black):,zzou:=spacecurve(0,0,z,z=a-1/2.b+1/2,thickness=3,color=black):,yuan1:=spacecurve(R*cos(t),R*sin(t),0,t=0.2*Pi,thickness=3,color=red):,yuan2:=spacecurve(R*cos(t),0,R*sin(t),t=0.2*Pi,thickness=3,color=red):,xi:=1*R:,zhengfangxing:=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):,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=a.xi,y=-f(x).f(x),color=green):,qumian2:=plot3d(x,y,-f(x),x=a.xi,y=-f(x).f(x),color=green):,qumian3:=plot3d(x,f(x),z,x=a.xi,z=-f(x).f(x),color=yellow):,qumian4:=plot3d(x,-f(x),z,x=a.xi,z=-f(x).f(x),color=yellow):,display(xzou,yzou,zzou,yuan1,yuan2,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,qumian4,scaling=constrained,orientation=60,73);,11,12,with(plots):,R:=1:,f:=x-sqrt(R2-x2):,a:=-R:b:=R:,xzou:=spacecurve(x,0,0,x=a-1.b+1,thickness=3,color=black):,yzou:=spacecurve(0,y,0,y=a-1.b+1,thickness=3,color=black):,zzou:=spacecurve(0,0,z,z=a-1.b+1,thickness=3,color=black):,zhumian1:=plot3d(R*cos(t),R*sin(t),z,t=0.2*Pi,z=a-1.b+1,style=wireframe,color=blue):,zhumian2:=plot3d(R*cos(t),y,R*sin(t),t=0.2*Pi,y=a-1.b+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=grey,style=patchnogrid):,qumian1i:=plot3d(x,y,f(x),x=a.xi,y=-f(x).f(x),color=green,style=patchnogrid):,qumian2i:=plot3d(x,y,-f(x),x=a.xi,y=-f(x).f(x),color=green,style=patchnogrid):,qumian3i:=plot3d(x,f(x),z,x=a.xi,z=-f(x).f(x),color=yellow,style=patchnogrid):,qumian4i:=plot3d(x,-f(x),z,x=a.xi,z=-f(x).f(x),color=yellow,style=patchnogrid)od:,zhengfangxing:=display(seq(zhengfangxingi,i=0.K),insequence=true):,zhengfangban:=display(seq(zhengfangbani,i=0.K),insequence=true):,qumian1:=display(seq(qumian1i,i=0.K),insequence=true):,qumian2:=display(seq(qumian2i,i=0.K),insequence=true):,qumian3:=display(seq(qumian3i,i=0.K),insequence=true):,qumian4:=display(seq(qumian4i,i=0.K),insequence=true):,display(xzou,yzou,zzou,zhumian1,zhumian2,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,qumian4,scaling=constrained);,动画的,Maple,程序,13,下面来求牟合方盖的体积,14,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：,15,牟合方盖,的体积,与下面这个立体的体积相等,16,17,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-1.b+1,thickness=3,color=black):,yzou:=spacecurve(0,y,0,y=a-1.b+1,thickness=3,color=black):,base:=plot3d(x,y,0,x=a.b,y=g(x).f(x),color=grey,style=patchnogrid):,quxian:=spacecurve(R*cos(t),R*sin(t),0,t=0.2*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=0.f(xi)-g(xi),color=yellow,style=patchnogrid):,qumian1i:=plot3d(x,f(x),(f(x)-g(x)*t,t=0.1,x=a.xi,color=green):,qumian2i:=plot3d(x,g(x),(f(x)-g(x)*t,t=0.1,x=a.xi,color=green):,qumian3i:=plot3d(x,g(x)+(f(x)-g(x)*t,f(x)-g(x),t=0.1,x=a.xi,color=grey)od:,zhengfangxing:=display(seq(zhengfangxingi,i=0.K),insequence=true):,zhengfangban:=display(seq(zhengfangbani,i=0.K),insequence=true):,qumian1:=display(seq(qumian1i,i=0.K),insequence=true):,qumian2:=display(seq(qumian2i,i=0.K),insequence=true):,qumian3:=display(seq(qumian3i,i=0.K),insequence=true):,display(xzou,yzou,base,quxian,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,scaling=constrained,orientation=-60,70);,动画的,Maple,程序,18,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,19,20,