用二重积分计算旋转体体积的几何解释

1、 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 1 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 2 设设D是上半平面内的一个有界闭区域。是上半平面内的一个有界闭区域。 将将D绕绕x轴轴旋转一周得一旋转体，求该旋旋转一周得一旋转体，求该旋转体的体积转体的体积V。 我们用元素法来建立旋转体体积的二我们用元素法来建立旋转体体积的二重积分公式。重积分公式。D July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 3d( , )x

2、yD在区域在区域D的的(x,y)处取一个面积元素处取一个面积元素d它到它到x轴的距离是轴的距离是 y （如图）。（如图）。该面积元素绕该面积元素绕x轴轴旋转而成的旋转体的体积约为：旋转而成的旋转体的体积约为：2dVyd（体积元素）（体积元素）于是整个区域绕于是整个区域绕x轴轴旋转而旋转而成的旋转体的体积为：成的旋转体的体积为：2DDVydVdy July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 4d( , )x yD命题命题1：上半平面内一个有界闭区域：上半平面内一个有界闭区域D绕绕x轴轴旋转而成的旋转体的体积为：旋转而成的旋转体的体积为

3、：2DdVyy July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 5下面来解释以上公式的几何意义下面来解释以上公式的几何意义 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 6区域区域D中一面积元素中一面积元素 绕绕x轴轴旋转而成旋转而成的旋转体为一的旋转体为一环形体环形体(如图如图)。dd July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 7区域区域D中一面积元素中一面积元素 绕绕x轴轴旋转而成旋转而成的旋转体为一环形体的旋

4、转体为一环形体(如图如图)。d其体积约为：其体积约为：2dVyd（体积元素）（体积元素） July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 8将将dV在在D上二重积分的几何意义是将划分上二重积分的几何意义是将划分D的的n个面积元素分别绕个面积元素分别绕x轴旋转而成的旋转体轴旋转而成的旋转体相加，得到整个相加，得到整个D绕绕x轴旋转的旋转体。轴旋转的旋转体。于是整个区域绕于是整个区域绕x轴轴旋转而成的旋转体旋转而成的旋转体的体积为：的体积为：2DDVydVd July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体

5、体积计算公式的几何意义 9以下图形给出了这种方法的几何解释以下图形给出了这种方法的几何解释 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 10 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 11display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4

6、_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constraine

7、d,color=green); July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 12 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28

8、,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 13 display(xzhou

9、,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y

10、_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 14 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 15设想用电缆做成一个圆环体设想用电缆做成一个圆环体那么这个圆环体可由电缆中很多圆环形那么

11、这个圆环体可由电缆中很多圆环形状的光纤组成状的光纤组成因此，我们可以把这种计算旋转体体积因此，我们可以把这种计算旋转体体积的方法形象地称为的方法形象地称为光纤法光纤法或或电缆法电缆法 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 16更多的图形更多的图形 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 17 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 18 July 4, 2012四川大学数学学院 徐小湛旋转体体积计

12、算公式的几何意义旋转体体积计算公式的几何意义 19with(plots):xzhou:=spacecurve(x,0,0, x=-2.2, thickness=1,color=black):yzhou:=spacecurve(0,y,0, y=-2.2, thickness=1,color=black):zzhou:=spacecurve(0,0,z, z=-2.4, thickness=1,color=black):a:=0:b:=3:R:=1:r:=0.1:yuan:=spacecurve(0,a+R*cos(t),b+R*sin(t), t=0.2*Pi, thickness=3,col

13、or=red):a0:=0:a2:=0.2:a4:=0.4:a6:=0.6:a8:=0.8:a_2:=-0.2:a_4:=-0.4:a_6:=-0.6:a_8:=-0.8:b0:=3:b2:=3.2:b4:=3.4:b6:=3.6:b8:=3.8:b_2:=3-0.2:b_4:=3-0.4:b_6:=3-0.6:b_8:=3-0.8:y00:=spacecurve(0,a0+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y04:

14、=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y06:=spacecurve(0,a0+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y08:=spacecurve(0,a0+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y0_2:=spacecurve(0,a0+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y0_4:=spacecurve(0,a0+r*cos(t),b_4+r*sin(t), t

15、=0.2*Pi,color=blue):y0_6:=spacecurve(0,a0+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y0_8:=spacecurve(0,a0+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y20:=spacecurve(0,a2+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y22:=spacecurve(0,a2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y24:=spacecurve(0,a2

16、+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y26:=spacecurve(0,a2+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y28:=spacecurve(0,a2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y2_2:=spacecurve(0,a2+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y2_4:=spacecurve(0,a2+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=bl

17、ue):y2_6:=spacecurve(0,a2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y2_8:=spacecurve(0,a2+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y40:=spacecurve(0,a4+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y42:=spacecurve(0,a4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y44:=spacecurve(0,a4+r*cos(t),b4+r*s

18、in(t), t=0.2*Pi,color=blue):y46:=spacecurve(0,a4+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y48:=spacecurve(0,a4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y4_2:=spacecurve(0,a4+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y4_4:=spacecurve(0,a4+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y4_6:=spacec

19、urve(0,a4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y4_8:=spacecurve(0,a4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y60:=spacecurve(0,a6+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y62:=spacecurve(0,a6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y64:=spacecurve(0,a6+r*cos(t),b4+r*sin(t), t=0.2*Pi,

20、color=blue):y66:=spacecurve(0,a6+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y6_2:=spacecurve(0,a6+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y6_4:=spacecurve(0,a6+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y6_6:=spacecurve(0,a6+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y80:=spacecurve(0,a8+r*cos

21、(t),b0+r*sin(t), t=0.2*Pi,color=blue):y82:=spacecurve(0,a8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y84:=spacecurve(0,a8+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y8_2:=spacecurve(0,a8+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y8_4:=spacecurve(0,a8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y8

22、_6:=spacecurve(0,a8+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_20:=spacecurve(0,a_2+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_22:=spacecurve(0,a_2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_24:=spacecurve(0,a_2+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_26:=spacecurve(0,a_2+r*cos(t),b6+r*s

23、in(t), t=0.2*Pi,color=blue):y_28:=spacecurve(0,a_2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_2_2:=spacecurve(0,a_2+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_2_4:=spacecurve(0,a_2+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_2_6:=spacecurve(0,a_2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_

24、2_8:=spacecurve(0,a_2+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_40:=spacecurve(0,a_4+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_42:=spacecurve(0,a_4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_44:=spacecurve(0,a_4+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_46:=spacecurve(0,a_4+r*cos(t),b6+r

25、*sin(t), t=0.2*Pi,color=blue):y_48:=spacecurve(0,a_4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_4_2:=spacecurve(0,a_4+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_4_4:=spacecurve(0,a_4+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_4_6:=spacecurve(0,a_4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):

26、y_4_8:=spacecurve(0,a_4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_60:=spacecurve(0,a_6+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_62:=spacecurve(0,a_6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_64:=spacecurve(0,a_6+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_66:=spacecurve(0,a_6+r*cos(t),b6

27、+r*sin(t), t=0.2*Pi,color=blue):y_6_2:=spacecurve(0,a_6+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_6_4:=spacecurve(0,a_6+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_6_6:=spacecurve(0,a_6+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_80:=spacecurve(0,a_8+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue

28、):y_82:=spacecurve(0,a_8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_84:=spacecurve(0,a_8+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_8_2:=spacecurve(0,a_8+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_8_4:=spacecurve(0,a_8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_8_6:=spacecurve(0,a_8+r*cos(

29、t),b_6+r*sin(t), t=0.2*Pi,color=blue):h00:=plot3d(b0+r*cos(t)*sin(u),a0+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h02:=plot3d(b2+r*cos(t)*sin(u),a0+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h04:=plot3d(b4+r*cos(t)*sin(u),a0+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h06:=plot3d(b6+r

30、*cos(t)*sin(u),a0+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h08:=plot3d(b8+r*cos(t)*sin(u),a0+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_2:=plot3d(b_2+r*cos(t)*sin(u),a0+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_4:=plot3d(b_4+r*cos(t)*sin(u),a0+r*sin(t),(b_4+r*cos(t)*cos(u),t

31、=0.2*Pi,u=0.2*Pi):h0_6:=plot3d(b_6+r*cos(t)*sin(u),a0+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_8:=plot3d(b_8+r*cos(t)*sin(u),a0+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h20:=plot3d(b0+r*cos(t)*sin(u),a2+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h22:=plot3d(b2+r*cos(t)*sin(u)

32、,a2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h24:=plot3d(b4+r*cos(t)*sin(u),a2+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h26:=plot3d(b6+r*cos(t)*sin(u),a2+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h28:=plot3d(b8+r*cos(t)*sin(u),a2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2

33、_2:=plot3d(b_2+r*cos(t)*sin(u),a2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_4:=plot3d(b_4+r*cos(t)*sin(u),a2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_6:=plot3d(b_6+r*cos(t)*sin(u),a2+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_8:=plot3d(b_8+r*cos(t)*sin(u),a2+r*sin(t),(b

34、_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h40:=plot3d(b0+r*cos(t)*sin(u),a4+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h42:=plot3d(b2+r*cos(t)*sin(u),a4+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h44:=plot3d(b4+r*cos(t)*sin(u),a4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h46:=plot3d(b6+r

35、*cos(t)*sin(u),a4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h48:=plot3d(b8+r*cos(t)*sin(u),a4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_2:=plot3d(b_2+r*cos(t)*sin(u),a4+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_4:=plot3d(b_4+r*cos(t)*sin(u),a4+r*sin(t),(b_4+r*cos(t)*cos(u),t

36、=0.2*Pi,u=0.2*Pi):h4_6:=plot3d(b_6+r*cos(t)*sin(u),a4+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_8:=plot3d(b_8+r*cos(t)*sin(u),a4+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h60:=plot3d(b0+r*cos(t)*sin(u),a6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h62:=plot3d(b2+r*cos(t)*sin(u)

37、,a6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h64:=plot3d(b4+r*cos(t)*sin(u),a6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h66:=plot3d(b6+r*cos(t)*sin(u),a6+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h6_2:=plot3d(b_2+r*cos(t)*sin(u),a6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi)

38、:h6_4:=plot3d(b_4+r*cos(t)*sin(u),a6+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h6_6:=plot3d(b_6+r*cos(t)*sin(u),a6+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h80:=plot3d(b0+r*cos(t)*sin(u),a8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h82:=plot3d(b2+r*cos(t)*sin(u),a8+r*sin(t),(b2+

39、r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h84:=plot3d(b4+r*cos(t)*sin(u),a8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h8_2:=plot3d(b_2+r*cos(t)*sin(u),a8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h8_4:=plot3d(b_4+r*cos(t)*sin(u),a8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_20:=plot3d(

40、b0+r*cos(t)*sin(u),a_2+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_22:=plot3d(b2+r*cos(t)*sin(u),a_2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_24:=plot3d(b4+r*cos(t)*sin(u),a_2+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_26:=plot3d(b6+r*cos(t)*sin(u),a_2+r*sin(t),(b6+r*cos(t)*cos

41、(u),t=0.2*Pi,u=0.2*Pi):h_28:=plot3d(b8+r*cos(t)*sin(u),a_2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_2:=plot3d(b_2+r*cos(t)*sin(u),a_2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_4:=plot3d(b_4+r*cos(t)*sin(u),a_2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_6:=plot3d(b_6+r

42、*cos(t)*sin(u),a_2+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_8:=plot3d(b_8+r*cos(t)*sin(u),a_2+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_40:=plot3d(b0+r*cos(t)*sin(u),a_4+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_42:=plot3d(b2+r*cos(t)*sin(u),a_4+r*sin(t),(b2+r*cos(t)*cos

43、(u),t=0.2*Pi,u=0.2*Pi):h_44:=plot3d(b4+r*cos(t)*sin(u),a_4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_46:=plot3d(b6+r*cos(t)*sin(u),a_4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_48:=plot3d(b8+r*cos(t)*sin(u),a_4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_2:=plot3d(b_2+r*cos(t

44、)*sin(u),a_4+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_4:=plot3d(b_4+r*cos(t)*sin(u),a_4+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_6:=plot3d(b_6+r*cos(t)*sin(u),a_4+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_8:=plot3d(b_8+r*cos(t)*sin(u),a_4+r*sin(t),(b_8+r*cos(t)*cos

45、(u),t=0.2*Pi,u=0.2*Pi):h_60:=plot3d(b0+r*cos(t)*sin(u),a_6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_62:=plot3d(b2+r*cos(t)*sin(u),a_6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_64:=plot3d(b4+r*cos(t)*sin(u),a_6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_66:=plot3d(b6+r*cos(t)*

46、sin(u),a_6+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_2:=plot3d(b_2+r*cos(t)*sin(u),a_6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_4:=plot3d(b_4+r*cos(t)*sin(u),a_6+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_6:=plot3d(b_6+r*cos(t)*sin(u),a_6+r*sin(t),(b_6+r*cos(t)*cos(u)

47、,t=0.2*Pi,u=0.2*Pi):h_80:=plot3d(b0+r*cos(t)*sin(u),a_8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_82:=plot3d(b2+r*cos(t)*sin(u),a_8+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_84:=plot3d(b4+r*cos(t)*sin(u),a_8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_8_2:=plot3d(b_2+r*cos(t)*s

48、in(u),a_8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_8_4:=plot3d(b_4+r*cos(t)*sin(u),a_8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60

49、,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h02,h04,h06,h08,h0_2,h0_4,h0_6,h0_8,h20,h22,h24,h26,h28,h2_2,h2_4,h2_6,h2_8,h40,h42

50、,h44,h46,h48,h4_2,h4_4,h4_6,h4_8,h60,h62,h64,h66,h6_2,h6_4,h6_6,h80,h82,h84,h8_2,h8_4,h_20,h_22,h_24,h_26,h_28,h_2_2,h_2_4,h_2_6,h_2_8,h_40,h_42,h_44,h_46,h_48,h_4_2,h_4_4,h_4_6,h_4_8,h_60,h_62,h_64,h_66,h_6_2,h_6_4,h_6_6,h_80,h_82,h_84,h_8_2,h_8_4,scaling=constrained); July 4, 2012四川大学数学学院 徐小湛旋转体体积

51、计算公式的几何意义旋转体体积计算公式的几何意义 20 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,

52、y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h04,h08,h0_4,h0_8,h22,h26,h2_2,h2_6,h44,h48,h4_4,h4_8,h62,h66,h6_2,h6_6,h80,h84,h8_4,h_20,h_24,h_28,h_2_4,h_2_8,h_44,h_48,h_4_4,h_4_8,h_62,h_66,h_6_2,h_6_6,h_80,h_84,h_8_4,scaling=constrained); July 4,

53、2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 21 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8

54、,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h08,h0_8,h26,h2_6,h44,h4_4,h62,h6_2,h80,h84,h8_4,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained); July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 22d

55、isplay(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8

56、,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 23display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,

57、y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,

58、h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 24 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 25 July 4, 2012四川大学数学学院 徐小湛旋转体体积计算公式的几何意义旋转体体积计算公式的几何意义 26 with(plots):xzhou:=spacecurve(x,0,0, x=-2.2, thickness=1,col

59、or=black):yzhou:=spacecurve(0,y,0, y=-2.2, thickness=1,color=black):zzhou:=spacecurve(0,0,z, z=-2.4, thickness=1,color=black):a:=0:b:=3:R:=1:r:=0.1:yuan:=spacecurve(0,a+R*cos(t),b+R*sin(t), t=0.2*Pi, thickness=3,color=red):a0:=0:a2:=0.2:a4:=0.4:a6:=0.6:a8:=0.8:a_2:=-0.2:a_4:=-0.4:a_6:=-0.6:a_8:=-0.8

60、:b0:=3:b2:=3.2:b4:=3.4:b6:=3.6:b8:=3.8:b_2:=3-0.2:b_4:=3-0.4:b_6:=3-0.6:b_8:=3-0.8:y00:=spacecurve(0,a0+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y04:=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y06:=spacecurve(0,a0+r*co