用切片法讨论牟合方盖

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 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 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 牟合方盖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 R 2 x 2 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 forifrom0toKdoxi a i b a K zhengfangxing i 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 i plot3d xi y z y f xi f xi z f xi f xi color grey style patchnogrid qumian1 i plot3d x y f x x a xi y f x f x color green qumian2 i plot3d x y f x x a xi y f x f x color green qumian3 i plot3d x f x z x a xi z f x f x color yellow qumian4 i plot3d x f x z x a xi z f x f x color yellow od zhengfangxing display seq zhengfangxing i i 0 K insequence true zhengfangban display seq zhengfangban i i 0 K insequence true qumian1 display seq qumian1 i i 0 K insequence true qumian2 display seq qumian2 i i 0 K insequence true qumian3 display seq qumian3 i i 0 K insequence true qumian4 display seq qumian4 i i 0 K 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 R 2 x 2 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 牟合方盖12 牟合方盖13 with plots R 1 f x sqrt R 2 x 2 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 forifrom0toKdoxi a i b a K zhengfangxing i 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 i plot3d xi y z y f xi f xi z f xi f xi color grey style patchnogrid qumian1 i plot3d x y f x x a xi y f x f x color green style patchnogrid qumian2 i plot3d x y f x x a xi y f x f x color green style patchnogrid qumian3 i plot3d x f x z x a xi z f x f x color yellow style patchnogrid qumian4 i plot3d x f x z x a xi z f x f x color yellow style patchnogrid od zhengfangxing display seq zhengfangxing i i 0 K insequence true zhengfangban display seq zhengfangban i i 0 K insequence true qumian1 display seq qumian1 i i 0 K insequence true qumian2 display seq qumian2 i i 0 K insequence true qumian3 display seq qumian3 i i 0 K insequence true qumian4 display seq qumian4 i 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 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 R 2 x 2 g x sqrt R 2 x 2 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 forifrom0toKdoxi a i b a K zhengfangxing i 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 zhengfangban i plot3d xi y z y g xi f xi z 0 f xi g xi color yellow style patchnogrid qumian1 i plot3d x f x f x g x t t 0 1 x a xi color green qumian2 i plot3d x g x f x g x t t 0 1 x a xi color green qumian3 i 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 zhengfangxing i i 0 K insequence true zhengfangban display seq zhengfangban i i 0 K insequence true qumian1 display seq qumian1 i i 0 K insequ