用切片法討論牟合方蓋.ppt

牟合方蓋 1,用切片法討論牟合方蓋,蜀南竹海,牟合方蓋 2,牟合方蓋就是兩個(gè)半徑相同的直交圓柱面所圍成的立體。 教材中一般是利用二重積分計(jì)算其體積。 本課件用截面的面積的定積分來計(jì)算其體積。 還用數(shù)學(xué)軟件Maple制作了有關(guān)動(dòng)畫。 最后比較牟合方蓋與另一個(gè)立體的體積。,牟合方蓋 3,求兩直交圓柱面,所圍成的立體的體積,牟合方蓋 4,牟合方蓋,牟合方蓋 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_axis:=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,thickness=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);,牟合方蓋 6,牟合方蓋,劉徽在他的九章算術(shù)注中，提出一個(gè)獨(dú)特的方法來計(jì)算球體的體積：他不直接求球體的體積，而是先計(jì)算另一個(gè)叫牟合方蓋的立體的體積。 所謂牟合方蓋，就是指由兩個(gè)同樣大小但軸心互相垂直的圓柱體相交而成的立體。由于這立體的外形似兩把上下對(duì)稱的正方形雨傘，所以就稱它為牟合方蓋。 在這個(gè)立體里面，可以內(nèi)切一個(gè)半徑和原本圓柱體一樣大小的球體，劉徽并指出，由于內(nèi)切圓的面積和外切正方形的面積之比為:4，所以牟合方蓋的體積與球體體積之比亦應(yīng)為:4。 可惜的是，劉徽并沒有求出牟合方蓋的體積，所以亦不知道球體體積的計(jì)算公式。,/subject/maths/printer.php?article_id=1258,牟合方蓋 7,下面用截面來研究牟合方蓋,牟合方蓋 8,牟合方蓋 9,從 x 軸正向看去,牟合方蓋 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): 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): 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=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),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),insequence=true): qumian4:=display(seq(qumian4i,i=0K),insequence=true): display(xzou,yzou,zzou,zhengfangban,zhengfangxing,qumian1,qumian2,qumian3,qumian4,scaling=constrained);,動(dòng)畫的Maple程序,牟合方蓋 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): yzou:=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(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): 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);,牟合方蓋 12,牟合方蓋 13,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): 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),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=grey,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),color=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,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);,動(dòng)畫的Maple程序,牟合方蓋 14,下面來求牟合方蓋的體積,牟合方蓋 15,這與用二重積分計(jì)算的結(jié)果相同 見同濟(jì)高等數(shù)學(xué)六版，下冊(cè) 143頁(yè)，例4,垂直于x軸的截面是一個(gè)正方形：,牟合方蓋 16,牟合方蓋的體積與下面這個(gè)立體的體積相等,牟合方蓋 17,牟合方蓋 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:=spacecurve(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,style=patchnogrid): qumian1i:=plot3d(x,f(x),(f(x)-g(x)*t,t=01,x=axi,color